Je Voudrais Aussi Mentionner

Romain Couillet

Mérouane Debbah

Walid Hachem

Abla Kammoun

Je Remercie

Charles Bordenave

Catherine Donati-Martin

Sandrine Péché

Amandine, Andrés, Charanpal, Charles Cristina Alexandre

Yu Ding

Éric, Joffrey, Marc, Mony, Nicolas Émilie K Olaf

Onur Sylvain L C Olivier

Xing Je Remercie Fabrice

Yu Chen

Lie Fu

Fangzhou Jin

Shinan Liu

Zhenkai Lv

Zhe Tong

Bin Xu

Fangjia Yan

Yuan Bian

Jiaming Chen

Fanding Gu

Yue Huang

Wenfeng Li

Wentian Liu

Yizuo Chen

Zhoujie Ding

Hui Jiang

Shuoyan Liu

Yuntao Ni

Songyan Xie

Xin Bien D'autres (

Bin He

Meilin Li

Yang Li

Yang Wang

Yixian Yang

Zhou

) Parmi

en outre un grand honneur pour moi d'avoir

Je tiens d'abord à remercier chaleureusement mon directeur de thèse Jamal Najim. Sa générosité constante, la richesse de ses intuitions et sa grande disponibilité ont rendu ces années très enrichissantes. J'ai appris non seulement les mathématiques, mais aussi la rigueur et la qualité de la rédaction. Il est bien évident que sans lui, cette thèse n'aurait pas vu le jour et je serais loin de profiter des mathématiques qu'il m'a transmises. Sa façon de travailler restera pour moi un modèle.

Je remercie sincèrement mon co-directeur de thèse Éric Moulines pour m'avoir proposé un sujet très riche et d'intégrer un laboratoire où l'ambiance est conviviale. Son aide a été chaleureuse et précieuse.

Résumé

L'objectif principal de la thèse est : l'étude des fluctuations de fonctionnelles du spectre de grandes matrices aléatoires, la construction d'estimateurs consistants et l'étude de leurs performances, dans la situation où la dimension des observations est du même ordre que le nombre des observations disponibles. Un tel contexte se rencontre très fréquemment en communications numériques, principal domaine applicatif de la thèse. On retrouve les mêmes problématiques dès lors que l'on traite des observations de très grande dimension (théorie de l'apprentissage, biostatistiques, etc.), ou que des contraintes fortes de temps réel (réduisant le nombre d'observations que l'on peut acquérir) entrent en jeu (radio cognitive, radio intelligente, réseaux de capteurs).

Il y aura deux grandes parties dans cette thèse. La première concerne la contribution méthodologique. Nous ferons l'étude des fluctuations pour les statistiques linéaires des valeurs propres du modèle 'information-plus-bruit' pour des fonctionnelles analytiques, et étendrons ces résultats au cas des fonctionnelles non analytiques. Le procédé d'extension est fondé sur des méthodes d'interpolation avec des quantités gaussiennes. Ce procédé est appliqué aux grandes matrices de covariance empirique. L'autre grande partie sera consacrée à l'estimation des valeurs propres de la vraie covariance à partir d'une matrice de covariance empirique en grande dimension et l'étude de son comportement. Nous proposons un nouvel estimateur consistant et étudions ces fluctuations. En communications sans fil, cette procédure permet à un réseau secondaire d'établir la présence de ressources spectrales disponibles.

Les objectifs scientifiques de la thèse présentée seront :

-Fluctuations de statistiques linéaires pour grandes matrices aléatoires, -Construction d'estimateurs consistants dans un contexte de traitement d'antenne et de canaux de propagation MIMO et l'étude de la performance de ces estimateurs.

Notations

Linear algebra X matrix

I N identity matrix of size N × N diag(x 1 , • • • , x N ) diagonal matrix with entries (x 1 , • • • , x N )
eig(X) the set of eigenvalues of X X ij ,[X] ij ,x ij (i, j) entry of matrix X X T transpose of matrix X X H or X * complex conjugate transpose of X X complex conjugate of X vdiag(X) vector with entries (X 11 , • • • , X N N )

Tr(X), det(X), rank(X) trace, determinant and rank of X X spectral norm of X x column vector

x i i-th entry of x ≤, ≥, >, < matrix inequalities, A ≥ B means A -B is nonnegative while A > B means A -B is nonnegative definite

Probability theory

(Ω, F, P) probability space with σ-algebra F and measure P E(X) expectation of X P probability F X distribution function of X, i.e., F X (x) = P(X ≤ x) supp(F) support of the distribution function F a.s.

--→ almost surely convergence

C * C\{0} i i = √ -1 with Im(i) = 1 |z| modulus of z I A indicator function (x n ) n≥1 sequence of x 1 , x 2 , • • • O(y n )
x n = O(y n ) means there exists C and n 0 such that x n ≤ Cy n for n ≥ n 0 o(y n )

x n = o(y n ) means for all ǫ > 0 and there exists n 0 such that

x n ≤ ǫy n for n ≥ n 0 f ′ (x),f ′′ (x), f (n) (x)
first, second and n-th derivative of f Chapitre 1

Introduction

L'origine de l'étude de matrices aléatoires remonte aux travaux du statisticien Wishart en 1928 [START_REF] Wishart | The generalized product moment distribution in samples from a normal multivariate population[END_REF] qui s'intéresse au comportement asymptotique de la matrice de covariance empirique définie par

R n = 1 n n i=1 x i x * i , où x i ∈ C N est i.i.d. centrée réduite.
En 1955 [START_REF] Wigner | Characteristic vectors of bordered matrices with infinite dimensions[END_REF], et pour des motivations de physique théorique [START_REF] Wigner | On the distribution of roots of certain symmetric matrices[END_REF], le physicien Wigner s'intéresse au comportement asymptotique du spectre d'une matrice aléatoire symétrique, à entrées indépendantes et correctement normalisées quand la dimension de la matrice tend vers l'infini. Wigner établit la convergence de la mesure empirique des valeurs propres vers la loi semi-circulaire.

Des résultats similaires voient le jour pour de grandes matrices de covariance empirique pour lesquelles les dimensions tendent vers l'infini au même rythme (rapport entre les dimensions constant). Le résultat emblématique est le théorème de Marčenko-Pastur [START_REF] Marcenko | Distributions of eigenvalues for some sets of random matrices[END_REF] qui décrit le comportement asymptotique de la mesure spectrale d'une grande matrice de covariance empirique ; s'en suivent l'étude des fluctuations de la plus grande valeur propre [START_REF] Tracy | On orthogonal and symplectic matrix ensembles[END_REF], l'étude des fluctuations du spectre [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF], certains théorèmes de grandes déviations [START_REF] Ben Arous | Erratum : "Large deviations for Langevin spin glass dynamics[END_REF], etc. D'un point de vue des applications, la théorie des grandes matrices aléatoires a suscité un grand intérêt en physique [START_REF] Mehta | Random matrices[END_REF], en biologie [START_REF] Arnold | On the asymptotic distribution of the eigenvalues of random matrices[END_REF], en finance [START_REF] Plerous | Random matrix approach to cross correlations in financial data[END_REF], en théorie des communications numériques [START_REF] Tse | Multiaccess fading channels. I. Polymatroid structure, optimal resource allocation and throughput capacities[END_REF][START_REF] Verdù | Spectral efficiency of CDMA with random spreading[END_REF][START_REF] Telatar | Capacity of multi-antenna Gaussian channels[END_REF][START_REF] Foschini | On limits of wireless communications in a fading environment when using multiple antennas[END_REF], etc.

Nos travaux ont porté sur l'estimation et l'étude des fluctuations de fonctionnelles de grandes matrices aléatoires de covariance empirique pour des modèles de matrices structurées : matrice de Gram et matrices du type information-plus-bruit. Dans cette introduction, nous présentons brièvement nos travaux ainsi que certains outils utilisés dans la suite de ce manuscrit.

CLT pour le modèle 'information-plus-bruit'

Considérons une matrice aléatoire Y n = (y n ij ) donnée par :

Y n = 1 √ n X n + A n ,
où X n est une matrice N × n dont les entrées (x n ij ; i, j, n) sont réelles ou complexes, i.i.d. de moyenne 0 et variance 1 ; la matrice A n a la même dimension et est déterministe. La matrice Y n est parfois appelée matrice du type "information-plus-bruit" dans la littérature.

Dans le Chapitre 2, nous étudierons les fluctuations de statistiques linéaires de la forme :

Trf (Y n Y * n ) = N i=1 f (λ i ) ,
où les λ i sont les valeurs propres de Y n Y * n , et f est une fonction analytique, dans le régime où les dimensions de n et N tendent vers l'infini à la même vitesse :

N, n → ∞ et 0 < lim inf N n ≤ lim sup N n < ∞ .
Les grandes matrices du type information-plus-bruit, et plus généralement, les grandes matrices aléatoires non centrées, ont récemment attiré l'attention de chercheurs comme dans Loubaton et al. [START_REF] Vallet | Improved subspace estimation for multivariate observations of high dimension : The deterministic signals case[END_REF], Hachem et al. [START_REF] Hachem | On bilinear forms based on the resolvent of large random matrices. to be published[END_REF], Capitaine et al. [START_REF] Capitaine | The largest eigenvalues of finite rank deformation of large Wigner matrices : convergence and nonuniversality of the fluctuations[END_REF]. Elles constituent un modèle important pour les applications. Sous des conditions peu restrictives sur les moments des entrées de la matrice X n et sur la norme spectrale de la matrice A n , le comportement asymptotique de la distribution empirique des valeurs propres de Y n Y * n :

F YnY * n = 1 N N i=1 δ λ i , (1.1.1)
a été étudié par Girko [START_REF] Girko | Theory of stochastic canonical equations[END_REF]Chapitre 7], Dozier et Silverstein [START_REF] Dozier | On the empirical distribution of eigenvalues of large dimensional information plus noise-type matrices[END_REF] et Hachem et al. [START_REF] Hachem | Deterministic Equivalents for Certain Functionals of Large Random Matrices[END_REF]. En suivant ces résultats, les propriétés variées du spectre asymptotique ont été étudiées, voir par exemple [START_REF] Dozier | Analysis of the Limiting Spectral Distribution of Large Dimensional Information-Plus-Noise Type Matrices[END_REF][START_REF] Loubaton | Almost sure localization of the eigenvalues in a Gaussian information plus noise model-application to the spiked models[END_REF][START_REF] Bai | No eigenvalues outside the support of the limiting spectral distribution of information-plus-noise type matrices[END_REF][START_REF] Capitaine | Exact separation phenomenon for the eigenvalues of large information-plusnoise type matrices. application to spiked models[END_REF]. Dans le contexte plus applicatif, les résultats ont été développés en communications numériques et traitement du signal [START_REF] Foschini | On limits of wireless communications in a fading environment when using multiple antennas[END_REF][START_REF] Telatar | Capacity of multi-antenna Gaussian channels[END_REF][START_REF] Moustakas | MIMO capacity through correlated channels in the presence of correlated interferers and noise : A (not so) large N analysis[END_REF][START_REF] Dumont | On the Capacity Achieving Covariance Matrix for Rician MIMO Channels : An Asymptotic Approach[END_REF][START_REF] Hachem | Large information plus noise random matrix models and consistent subspace estimation in large sensor networks[END_REF][START_REF] Hachem | A subspace estimator for fixed rank perturbations of large random matrices[END_REF]. Concernant les fluctuations de fonctionnelles du spectre de grandes matrices aléatoires, les références se retrouvent dans les monographies [START_REF] Anderson | An introduction to random matrices[END_REF][START_REF] Bai | Spectral analysis of large dimensional random matrices[END_REF][START_REF] Pastur | Eigenvalue distribution of large random matrices[END_REF][START_REF] Tao | Topics in random matrix theory[END_REF]. Dans le cas spécifique de fluctuations des grandes matrices de covariance, nous pouvons citer Jonsson [START_REF] Jonsson | Some limit theorems for the eigenvalues of a sample covariance matrix[END_REF], Johansson [START_REF] Johansson | On fluctuations of eigenvalues of random Hermitian matrices[END_REF], Cabanal-Duvillard [START_REF] Cabanal-Duvillard | Fluctuations de la loi empirique de grandes matrices aléatoires[END_REF], Guionnet [START_REF] Guionnet | Large deviations upper bounds and central limit theorems for noncommutative functionals of Gaussian large random matrices[END_REF], Bai et Silverstein [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF], Hachem et al. [START_REF] Hachem | A CLT for Information Theoretic Statistics of Gram Random Matrices with a Given Variance Profile[END_REF][START_REF] Hachem | A CLT for informationtheoretic statistics of non-centered Gram random matrices[END_REF], Pan et al. [START_REF] Pan | Central limit theorem for signal-to-interference ratio of reduced rank linear receiver[END_REF][START_REF] Bao | On the MIMO channel capacity for the general channels[END_REF], Chatterjee [START_REF] Chatterjee | Fluctuations of eigenvalues and second order Poincaré inequalities[END_REF], Lytova et Pastur [START_REF] Lytova | Central Limit Theorem for linear eigenvalue statistics of random matrices with independent entries[END_REF], Shcherbina [START_REF] Pastur | Eigenvalue distribution of large random matrices[END_REF], Najim [START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF], Khorunzhiy et al. [START_REF] Hachem | A new approach for mutual information analysis of large dimensional multi-antenna channels[END_REF], Benaych-Georges et al. [START_REF] Benaych-George | Central limit theorems for linear statistics of heavy tailed random matrices[END_REF], mais beaucoup d'autres articles méritent d'être cités.

Dans le cas de matrices non centrées du type information-plus-bruit, peu de résultats ont été développés concernant leurs fluctuations. Dans le cas spécifique où la fonctionnelle d'intérêt est

log det (I n + Y n Y * n ) = N i=1 log(1 + λ i ) ,
les fluctuations ont été décrites à l'aide de techniques physiciennes [START_REF] Moustakas | MIMO capacity through correlated channels in the presence of correlated interferers and noise : a (not so) large N analysis[END_REF] dans le cas où les entrées sont gaussiennes complexes, et le CLT a été établi dans le cas général par Hachem et al. dans [START_REF] Hachem | A CLT for informationtheoretic statistics of non-centered Gram random matrices[END_REF]. Notre étude portera sur les fluctuations de Trf (Y n Y * n ) dans le cas où f est analytique. Dans ce cas, la formule de Cauchy entraîne que .

Trf (Y n Y * n ) = - 1 2iπ C f (z) Tr (Y n Y * n -zI N ) -1 d z . (1.1.2) Soit Q n (z) = (Y n Y * n -zI N ) -1 (resp. Qn (z) = (Y * n Y n -zI n ) -1 ) la résolvante de Y n Y * n (resp. Y * n Y n ). Considérons δ n , δn : C + → C + , qui satisfont    δ n (z) = 1 n Tr -z(1 + δn )I N + AnA * n 1+δn -1 δn (z) = 1 n Tr -z(1 + δ n )I n + A * n An 1+ δn -1 , ( 1
(1.1.4)

Dans [START_REF] Hachem | On bilinear forms based on the resolvent of large random matrices. to be published[END_REF], ils démontrent que n N δ n (resp. δn (z)) est la transformée de Stieljes d'une mesure de probabilité F n (resp. Fn ) et T n (resp. Tn ) est une bonne approximation de Q n (resp. Qn ) dans le sens où

1 n TrT n (z) - 1 n TrQ n (z) p.s.
-----→

N,n→∞ 0, u * Q n v -u * T n v p.s.
-----→ N,n→∞ 0, où u, v sont deux vecteurs N × 1 avec la norme bornée. La statistique linéaire qui nous intéresse peut s'écrire sous la forme

M n (z) △ = TrQ n (z) -TrT n (z).
(1.1.5)

On peut décomposer (1.1.5) en deux parties :

Tr(Q n (z) -T n (z)) = Tr(Q n (z) -E Q n (z)) Fluctuations + Tr(E Q n (z) -T n (z)) Biais △ = M 1 n (z) + M 2 n (z) .
Le premier terme, qui est centré, donnera lieu aux fluctuations de la statistique linéaire et sera géré par la méthode des martingales 1 . Cette stratégie a été appliquée avec succès dans [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF][START_REF] Pan | Central limit theorem for signal-to-interference ratio of reduced rank linear receiver[END_REF][START_REF] Hachem | A CLT for Information Theoretic Statistics of Gram Random Matrices with a Given Variance Profile[END_REF][START_REF] Kammoun | A central limit theorem for the sinr at the lmmse estimator output for large-dimensional signals[END_REF][START_REF] Hachem | Large information plus noise random matrix models and consistent subspace estimation in large sensor networks[END_REF][START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF][START_REF] Bao | On the MIMO channel capacity for the general channels[END_REF]. Le second terme est complètement déterministe et donnera le biais. Dans le cas où A n = 0, nous retrouverons l'expression de la variance dans [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF] et du biais dans [START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF] (cf. Section 2.2.3). Trois difficultés majeures apparaissent dans la preuve :

1. Comme nous travaillons avec des variables générales ayant un 4 e moment fini, deux termes supplémentaires apparaîtrons dans l'étude de la variance asymptotique : un terme proportionnel au cumulant d'ordre 4, et un terme proportionnel à E(x n 11 ) 2 . Naturellement, ces termes dépendent de la matrice A n A * n . Mais malheureusement, ils ne dépendent pas uniquement du spectre de A n A * n , mais aussi des vecteurs propres de A n A * n . Cela entraîne qu'il est difficile de formuler des hypothèses naturelles sur A n A * n permettant la convergence de ces deux termes. Au lieu d'exprimer le résultat sous la forme habituelle

Trf (Y n Y * n ) -N f (x)F n (dx) D -----→ N,n→∞ N (B ∞ , Θ ∞ ),
on l'exprimera sous la forme

d LP Trf (Y n Y * n ) -N f (x)F n (dx), N (B n , Θ n ) -----→ N,n→∞ 0
à l'aide de la distance de Lévy-Prohorov entre la statistique linéaire d'intérêt et une famille de variables gaussiennes, qui s'ajustera à la famille des matrices. Cela permettra d'éviter de formuler des hypothèses peu naturelles sur la convergence jointe des vecteurs propres et du spectre de A n A * n . (cf. Section 2.2) 2. L'introduction d'une famille de gaussiennes N (B n , Θ n ) rend nécessaire différents contrôles asymptotiques, par exemple, la tension de la famille gaussienne, une borne uniforme de la variance et du biais. Malheureusement, la définition de leurs paramètres B n et Θ n repose sur les équations implicites définissant δ n et δn (cf. Eq (1.1.3)). Pour contourner ce problème, nous introduisons un méta-modèle qui permet de transférer les propriétés de notre statistique linéaire à cette famille de gaussiennes. Le méta-modèle consiste à introduire un nouveau paramètre M ∈ N pour que δ n soit la limite de la résolvante normalisée associée à la matrice définie par 3. Le calcul de la covariance s'inspire fortement des calculs faits dans Hachem et al. [START_REF] Hachem | A CLT for informationtheoretic statistics of non-centered Gram random matrices[END_REF] dans le cadre particulier de la fonctionnelle log det(I N + Y n Y * n ). Du fait de la formule de dérivation

A n (M ) =     A n • • • 0 . . . . . . . . . 0 • • • A n     et Y n (M ) = 1 √ nM X n (M ) + A n (M ),
∂ ∂ρ log det (ρI n + Y n Y * n ) = N i=1 1 ρ + λ i , pour ρ > 0,
il existe des liens profonds avec les calculs que nous devons mener. Une différence importante réside sur le fait que nous travaillons dans le plan complexe alors que Hachem et al. travaillent sur l'axe réel. Cette différence induit des difficultés techniques substantielles (cf. Section 2.4).

Fluctuations pour des fonctionnelles non analytiques

Dans le Chapitre 3, nous nous proposons d'étudier les fluctuations des fonctionnelles nonanalytiques pour le modèle

Y n = 1 √ n R 1/2
n X n où les entrées sont supposées réelles, centrées réduites et R n est une matrice symmétrique avec la norme spectrale bornée. La représentation (1.1.2) est la clé pour la méthode des martingales et impose une contrainte très forte sur la nature de la fonction f , qui doit être analytique sur un contour spécifique.

Afin d'affaiblir l'hypothèse d'analycité de f , nous allons procéder différemment et adopter le point de vue de Lytova et Pastur [START_REF] Lytova | Central Limit Theorem for linear eigenvalue statistics of random matrices with independent entries[END_REF]. Pour des entrées réelles, une autre manière de décomposer une fonction est la transformée de Fourier : f (t) = (2π) -1 e -itλ f (λ) dλ et sa formule d'inversion f (λ) = e iλt f (t) dt, qui amène la représentation :

Trf (Y n Y T n ) = f (t) Tre itYnY T n dt . (1.2.1)
Nous pouvons remarquer que la représentation précédente ne nécessite pas que f soit analytique, et c'est là l'un des grands avantages pour décomposer une fonction dans le domaine de Fourier plutôt que par la formule de Cauchy (dans le domaine de Stieljes).

La stratégie est maintenant claire : La fonction z → e itz étant analytique sur C, nous pouvons décrire les fluctuations de

Tr[e itYnY T n ] = - 1 2iπ e itz Tr Y n Y T n -zI N -1 d z ,
à l'aide de l'extension du théorème de Bai et Silverstein comme développé dans [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF], et un argument de tension va fournir un CLT pour

K -K f (t) Tr[e itYnY T n ] dt. (1.2.2)
Pour retrouver la représentation (1.2.1), nous avons besoin d'étendre le CLT précédent sur l'axe réel (K → ∞). L'ingrédient clé sera un contrôle, à priori, de la variance de la variable aléatoire u n (t) = Tr[e itYnY T n ] qui sera un polynôme de t. Ce contrôle utilise les techniques d'interpolation développées par Lytova et Pastur [START_REF] Lytova | Central Limit Theorem for linear eigenvalue statistics of random matrices with independent entries[END_REF]. L'idée principale consiste à interpoler entre les statistiques linéaires u(t) = Tr[e itYnY T n ] et les statistiques linéaires gaussiennes réelles û(t) = Tr[e it Ŷn ŶT n ] en considérant la matrice d'interpolation

X s = √ sX n + √ 1 -s Xn ,
et les quantités correspondantes. Dans le cas gaussien, une application directe de l'inégalité de Nash-Poincaré [START_REF] Hachem | A new approach for capacity analysis of large dimensional multi-antenna channels[END_REF][START_REF] Pastur | A simple approach to global regime of random matrix theory[END_REF][START_REF] Capitaine | The largest eigenvalues of finite rank deformation of large Wigner matrices : convergence and nonuniversality of the fluctuations[END_REF]] aboutit à :

Var{û(t)} ≤ Kt 2 .

Les techniques d'interpolation nous permettent d'obtenir le contrôle suivant

Var[Tre itYnY T n ] ≤ K(1 + |t| 4 ) 2 .

(1.2.3)

Nos travaux sont constitués de 3 étapes :

1. Adapter les techniques d'interpolation de Lytova et Pastur au modèle

Y n = 1 √ n R 1/2 n X n + A n
où les normes spectrales de deux matrices déterministes A n et R n sont supposées bornées. L'ajout de la matrice A n ne présente pas de difficulté supplémentaires, aussi nous établissons la borne (1.2.3) pour ce modèle général (cf. Section 3.2.4).

Dans le reste du chapitre, nous considérerons A n = 0. En prenant R n = I n , cette borne permettrait d'appliquer cette technique au modèle information-plus-bruit étudié au Chapitre 2.

Établir le CLT au modèle

Y n = R 1/2
n X n avec des entrées réelles pour Trf (Y n Y T n ). Par la représentation (1.2.1), l'estimation de Tr[e itYnY T n ] aboutit à l'inégalité

Var 1/2 [ f (t)Tre itYnY T n ] ≤ K| f (t)|(1 + |t| 4 )
qui est par hypothèse, intégrable sur R. Un théorème de transfert [START_REF] Kallenberg | Foundations of mordern Probability[END_REF]Theorem 4.28] nous assure la convergence de l'intégrale (1.2.2) vers une gaussienne quand K → ∞ (cf. Section 3.3).

3. Étudier le biais E f (λ)F YnY T n (dλ) -N f (λ)F n (dλ) où F n est la measure associée à l'équivalent déterministe T n (z). Une approche directe du biais semble compliquée, aussi nous introduisons un intermédiare de calcul gaussien

E f (λ)F YnY T n (dλ) -N f (λ)F n (dλ) = E f (λ)F YnY T n (dλ) -E f (λ)F Ŷc Ŷ * c (dλ) + E f (λ)F Ŷc Ŷ * c (dλ) -N f (λ)F n (dλ) , (1.2.4
)

où Ŷc = 1 √ n R 1/2
n Xc et Xc est une matrice gaussienne complexe standard. Le premier terme se gère sans difficulté à l'aide de techniques d'interpolation. Pour le second terme, par un calcul gaussien développé dans [START_REF] Hachem | On bilinear forms based on the resolvent of large random matrices. to be published[END_REF][START_REF] Vallet | Improved subspace estimation for multivariate observations of high dimension : The deterministic signals case[END_REF], il existe deux polynômes P 12 et P 17 tels que

|ETrQ c (z) -TrT(z)| ≤ 1 n P 17 (|Imz| -1 )P 12 (|z|) (1.2.5)
où Q c est la résolvante associée à Ŷc Ŷ * c et P k désigne un polynôme de coefficients strictement positifs de degré k. Pour une fonction régulière (par exemple, de classe C 18 ), en utilisant la représentation

N E f (λ)F Ŷc Ŷ * c (dλ)-N f (λ)F n (dλ) = 1 π lim y↓0 R
f (x)Im{TrQ c (x+iy)-TrT(x+iy)}dx, la méthode de Haagerup et Thorbjørnsen [START_REF] Haagerup | A new application of random matrices : Ext(C * red (F 2 )) is not a group[END_REF] assure que,

N E f (λ)F Ŷc Ŷ * c (dλ) -N f (λ)F n (dλ) ≤ 1 n |(1 + D) p f (x)| P 12 (|x|) |Imz| 18-p dx,
où l'opérateur D = d dx . Autrement dit, la régularité de f compense l'ordre de la singularité de |Imz| -1 près de l'axe réel (cf. Section 3.4).

En ce qui concerne l'étude des fluctuations de

Trf (Y n Y T n ) -ETrf (Y n Y T n )
, nos hypothèses sur f sont comparables à celles de Lytova et Pastur [START_REF] Lytova | Central Limit Theorem for linear eigenvalue statistics of random matrices with independent entries[END_REF] et Shcherbina [START_REF] Shcherbina | Central limit theorem for linear eigenvalue statistics of the wigner and sample covariance random matrices[END_REF] sur plusieurs aspects. En effet, nous imposons

R | f (t)|(1 + |t| 4 )dt < ∞,
alors que la condition dans Pastur et Lytova [START_REF] Lytova | Central Limit Theorem for linear eigenvalue statistics of random matrices with independent entries[END_REF] est

R (1 + |t| 5 )| f (t)|dt < ∞,
et celle dans Shcherbina [START_REF] Shcherbina | Central limit theorem for linear eigenvalue statistics of the wigner and sample covariance random matrices[END_REF] est

R (1 + |t|) 3+ε | f (t)| 2 dt < ∞,
pour les entrées qui ont le moment d'ordre 4 + ε fini.

En ce qui concerne l'étude du biais, l'hypothèse sur f est beaucoup plus forte (f de classe C 18 ). Cela est compatible avec les travaux de Haagerup et Thorbjørnsen [START_REF] Haagerup | A new application of random matrices : Ext(C * red (F 2 )) is not a group[END_REF] (cas de Wigner gaussien complexe, où f est de classe C 8 ) et ceux de Schultz [START_REF] Schultz | Non-commutative polynomials of independent Gaussian random matrices. The real and symplectic cases[END_REF] (cas de Wigner réel, où f est de classe C 14 ). Il semble que la structure de notre modèle nécessite une hypothèse plus forte. La différence est due au contrôle de la variable gaussienne (cf. (1.2.5)).

Estimation en grande dimension

Dans le cas où on traite des données de grande dimension, c'est-à-dire, lorsque la dimension des données est du même ordre que la taille de l'échantillon, les méthodes classiques comme par exemple l'approximation de la matrice de covariance des observations par la matrice de covariance empirique échouent au motif que la structure de bruit dans un tel objet de grande dimension est trop importante pour qu'une moyennisation la fasse disparaître. Dans ce cadre-là, la construction d'estimateurs consistants est délicate mais des tendances se dégagent :

-il est illusoire de vouloir estimer de manière consistante des objets de grande taille M × M tels que la matrice de covariance de notre modèle, -il est possible d'estimer de manière consistante des fonctions de tels objets au motif que, moralement, le fait d'estimer une fonction d'une grande matrice et non la grande matrice elle-même opère une réduction drastique de la dimension de l'objet à estimer. Deux ingrédients majeurs interviennent dans l'estimation consistante de fonctions de grandes matrices :

-la compréhension du comportement asymptotique de la fonction à estimer. Dans le cadre de la théorie des matrices aléatoires, cette fonction dépend en général d'une matrice aléatoire dont le comportement asymptotique est traduit par des équations déterministes associées au modèle étudié (cf. [START_REF] Girko | Theory of stochastic canonical equations[END_REF], et [START_REF] Hachem | Deterministic Equivalents for Certain Functionals of Large Random Matrices[END_REF] pour les équations régissant le modèle de Rice corrélé par exemple). Cela a plusieurs conséquences pratiques importantes : la construction de l'estimateur d'une fonction donnée, par exemple, transformée de Stieltjes de la mesure spectrale d'une matrice de covariance donnée, dépendra de la fonction en question mais aussi des équations déterministes relatives au modèle matriciel considéré (ex : canal de Rayleigh, de Rice, etc.) ; autrement dit, la construction d'estimateurs se fait au cas par cas. -le contrôle local du spectre de matrices aléatoires (propriétés de séparation spectrale) permettant d'assurer la consistance des estimateurs construits. Pour certains types de canaux et certains modèle de grandes matrices aléatoires, les travaux théoriques [START_REF] Bai | No Eigenvalues Outside the Support of the Limiting Spectral Distribution of Large Dimensional Sample Covariance Matrices[END_REF][START_REF] Bai | Exact Separation of Eigenvalues of Large Dimensional Sample Covariance Matrices[END_REF] ont précisé les constructions de séparation spectrale nécessaires pour la construction d'estimateurs consistants [START_REF] Mestre | On the asymptotic behavior of the sample estimates of eigenvalues and eigenvectors of covariance matrices[END_REF][START_REF] Vallet | A G-estimator of the MIMO channel ergodic capacity[END_REF][START_REF] Vallet | Improved subspace doa estimation with large arrays : The deterministic signal case[END_REF].

Présentation de la problématique dans un cas particulier

Nous allons dans ce paragraphe présenter un exemple de situation permettant de bien comprendre la nature des problèmes qu'il convient de résoudre. Considérons un signal aléatoire (y j ) j∈Z de dimension N supposé observé entre les instants 1 et n. On suppose que les vecteurs (y j ) j=1,••• ,n sont de moyenne nulle, indépendants et de même loi. Désignons par R N la matrice de covariance des vecteurs (y j ) j=1,••• ,n .

Supposons que l'on souhaite estimer un paramètre scalaire dépendant de R N , par exemple Ceci provient évidemment du fait que RN -R N tend presque surement vers 0 dans ce régime (N fixé, n → ∞). En pratique, ceci signifie que si n est beaucoup plus grand que N , alors il est raisonnable d'estimer 1 N Tr(R N + xI N ) -1 par 1 N Tr( RN + xI N ) -1 . Il existe cependant des situations d'intérêt pratique dans lesquelles n, tout en restant supérieur à N , est du même ordre de grandeur que N . Afin d'évaluer la pertinence de l'estimateur dans ce cas, il est naturel d'étudier son comportement lorsque n et N tendent vers +∞ de telle sorte que le rapport N n → c, où 0 < c < 1. Dans ce contexte, la matrice RN est très loin de se comporter comme R N , et il n'y a donc aucune raison pour que 1 N Tr( RN + xI N ) -1 soit un estimateur consistant de 1 N Tr(R N + xI N ) -1 . Les matrices aléatoires telles que RN ont fait l'objet de nombreux travaux. En particulier, il a été établi par que la distribution des valeurs propres de RN converge vers une distribution déterministe dépendant de R N [START_REF] Silverstein | On the empirical distribution of eigenvalues of a class of large dimensional random matrices[END_REF]. Afin d'aboutir à ce résultat, il convient d'étudier la résolvante de RN définie pour tout x ≥ 0 comme la matrice ( RN + xI N ) -1 .

En traitement statistique du signal, l'estimateur proposé par X. Mestre [START_REF] Mestre | On the asymptotic behavior of the sample estimates of eigenvalues and eigenvectors of covariance matrices[END_REF][START_REF] Mestre | Improved estimation of eigenvalues of covariance matrices and their associated subspaces using their sample estimates[END_REF] repose sur une intégrale de contours. Considérons le modèle

Y n = 1 √ n R 1/2 n X n où la mesure spectrale de R n est supposée F Rn = L k=1 N k N δ λ k , L fixé et connu.
La condition de séparabilité suppose que les valeurs propres λ k sont suffisamment éloignées pour qu'il y ait L compacts disjoints dans la loi limite (voir Section 4.1.1 pour plus de détails). Sous cette condition, λ k peut s'écrire comme

λ k = - 1 2iπN k C k N i=1 1 λ i -z dz où le contour C k ne contient que λ k . Par le changement de variable z = -1 δn(z)
, λ k peut s'exprimer comme

λ k = N 2iπN k R k z δ′ n (z) δn (z) dz
où le contour C k ne contient que le k e compact. L'estimateur consistant est construit par le remplacement de l'equivalent déterministe δn (z) par la trace normalisée de la résolvante associée

mn (z) = 1 n Tr(Y * n Y n -zI n ) -1
Y n -zI n ) -1 et sa dérivée Tr(Y * n Y n -zI n ) -2
par le théorème de Bai et Silverstein [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF]. Un argument de tension permet de conclure le CLT. Nous proposons également un estimateur consistant de la variance. La seconde partie est consacrée à l'estimation sans condition de séparabilité par la méthode des moments et l'étude de son comportement.

Les points délicats sont :

1. Les propriétés sur le spectre de grandes matrices aléatoires sont en général faites pour la plus grande ou plus petite valeur propre. En revanche, il y a peu de résultats sur les positions de valeurs propres en dehors du contour R k . Dans [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF], pour tout k ∈ N, et µ r (resp. µ ℓ ), plus grand (resp. petit) que la plus grande (resp. petite) valeur du support S n de l'équivalent déterministe Fn , alors

P(λ YnY * n max > µ r ) = o(n -k ), P(λ YnY * n min < µ ℓ ) = o(n -k ).
Par l'inégalité de Nash-Poincaré, nous allons montrer que

P(max i d(λ YnY * n i , S n ) > ε) = o(n -k ), pour k ∈ N, où d(a, S n ) est la distance euclidienne entre a et S n (cf. Proposition 4.1.1).
2. L'estimation de la variance demande un calcul de résidu soigneux puisqu'il s'agit de calculer Fluctuations for linear spectral statistics of information-plus-noise type random matrices

R k R ′ ℓ m′ n (z 1 ) m′ n (z 2 ) ( mn (z 1 ) -mn (z 2 )) 2 - 1 (z 1 -z 2 ) 2 1 mn (z 1 ) mn (z 2 ) d z 1 d z 2 où mn (z) = 1 n Tr(Y * n Y n -zI n ) -1 (cf.
This chapter is inspired from [START_REF] Najim | Fluctuations for linear spectral statistics of large random informationplusnoise type matrices[END_REF]. Consider an N × n random matrix Y n = (y n ij ) given by :

Y n = 1 √ n X n + A n , (2.0.1)
where X n is an N ×n matrix whose entries (x n ij ; i, j, n) are real or complex, i.i.d. with mean 0 and variance 1 ; matrix A n has the same dimensions and is deterministic. Matrix Y n is sometimes coined as "Information-plus-noise" type matrix in the literature.

The purpose of this chapter is to study the fluctuations of linear spectral statistics of the form :

Trf (Y n Y * n ) = N i=1 f (λ i ) , (2.0.2)
where the λ i 's are the eigenvalues of Y n Y * n , and f is an analytic function, under the regime where the dimensions n and N = N (n) go to infinity at the same pace :

N, n → ∞ and 0 < lim inf N n ≤ lim sup N n < ∞ . (2.0.3)
This condition will simply be referred to as N, n → ∞ in the sequel.

Introduction

Resolvent, canonical equations and deterministic equivalents

Denote by Q n (z) and Qn (z) the resolvents of matrices

Y n Y * n and Y * n Y n : Q n (z) = (Y n Y * n -zI N ) -1 , Qn (z) = (Y * n Y n -zI n ) -1 , ( 2
m YnY * n (z) = 1 N Tr Q n (z) , m Y * n Yn (z) = 1 n Tr Qn (z) .
The following canonical equations admit a unique solution (δ, δ) in the class of Stieltjes transforms of nonnegative measures with support in R + (see for instance [START_REF] Hachem | Deterministic Equivalents for Certain Functionals of Large Random Matrices[END_REF][START_REF] Hachem | A CLT for informationtheoretic statistics of non-centered Gram random matrices[END_REF], see also [START_REF] Girko | Theory of stochastic canonical equations[END_REF]Section 7.11]) :

   δ n (z) = 1 n Tr -z(1 + δn )I N + AnA * n 1+δn(z) -1 δn (z) = 1 n Tr -z(1 + δ n )I n + A * n An 1+ δn(z) -1 . (2.1.2)
Moreover, n N δ n and δn are Stieltjes transforms of probability measures, i.e. there exist probability measures F n and Fn over R + such that :

n N δ n (z) = R + F n (dλ) λ -z and δn (z) = R + Fn (dλ) λ -z . (2.1.3)
Functions δ and δ being introduced, we can now define the following N × N and n × n matrices :

   T n (z) = -z(1 + δn (z))I N + AnA * n 1+δn(z) -1 Tn (z) = -z(1 + δ n (z))I n + A * n An 1+ δn(z) -1
.

(2.1.4)

Matrices T n and Tn can be thought of as deterministic equivalents of the resolvents Q n and Qn in the sense that they approximate the resolvents in various ways. For instance, for z ∈ C\R + :

1 n Tr(Q n (z) -T n (z))
a.s.

-----→ N,n→∞ 0 and 1 n Tr( Qn (z) -Tn (z))

a.s.

-----→ N,n→∞ 0 .

Interestingly, not only T n and Tn convey information on the limiting spectrum of the resolvents but also on their eigenvectors. It has been proved indeed in [START_REF] Hachem | On bilinear forms based on the resolvent of large random matrices. to be published[END_REF] that

u * n Q n v n -u * n T n v n -----→ N,n→∞ 0
(and a symmetric result for Qn and Tn ) where u n and v n are deterministic N × 1 vectors with uniformly bounded euclidian norms (in N ).

Representation of linear spectral statistics

We can now properly center the linear spectral statistics and introduce the main object of which we shall study the fluctuations :

L n (f ) = N i=1 f (λ i ) -N f (x) F n (dx)
as N, n → ∞, where F n is defined by (2.1.3). In the case where function f is analytic in a neighborhood of the limiting support of the spectrum of Y n Y * n (to be properly defined), Cauchy's integral formula yields (denote by Γ a contour surrounding this limiting spectrum on which f is analytic) :

L n (f ) = - 1 2iπ Γ f (z)Tr(Q n (z) -T n (z)) dz , (2.1.5)
which follows from the identities :

N i=1 f (λ i ) = - 1 2iπ Γ f (z)Tr Q n (z)dz , N f (x) F n (dx) = - 1 2iπ Γ f (z)Tr T n (z)dz ,
the last equality being an immediate consequence of (2.1.3), Cauchy's integral formula and Fubini's theorem.

Fluctuations and bias

It is clear from (2.1.5) that in order to describe the fluctuations of L n (f ), a good approach is to study the fluctuations of the process (Tr(Q n (z) -T n (z)) ; z ∈ Γ)). In order to proceed, we split

M n (z) △ = TrQ n (z) -TrT n (z) (2.1.6)
into two terms :

Tr(Q n (z) -T n (z)) = Tr(Q n (z) -E Q n (z)) Fluctuations + Tr(E Q n (z) -T n (z)) Bias △ = M 1 n (z) + M 2 n (z) .
The first term, which is centered, will give rise to the fluctuations of the linear spectral statistics and will be handled by martingale techniques. This strategy has been successfully applied in [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF][START_REF] Pan | Central limit theorem for signal-to-interference ratio of reduced rank linear receiver[END_REF][START_REF] Hachem | A CLT for Information Theoretic Statistics of Gram Random Matrices with a Given Variance Profile[END_REF][START_REF] Kammoun | A central limit theorem for the sinr at the lmmse estimator output for large-dimensional signals[END_REF][START_REF] Hachem | A CLT for informationtheoretic statistics of non-centered Gram random matrices[END_REF][START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF][START_REF] Bao | On the MIMO channel capacity for the general channels[END_REF] and will be followed. The second term is completely deterministic and will give rise to a bias.

Entries with non-null fourth cumulant and a family of gaussian random variables

It is well known since the paper by Khorunzhiy et al. [START_REF] Khorunzhy | Asymptotic properties of large random matrices with independent entries[END_REF] that if the fourth moment of the entries differs from the fourth moment of a Gaussian random variable, then a term appears in the variance of the trace of the resolvent, which is proportional to the fourth cumulant κ of the entries :

κ = E |x n 11 | 4 -|ϑ| 2 -2 where |ϑ| 2 = |E(x n 11 ) 2 | 2 .
The same phenomenon will occur here but the convergence of this additional term may fail to happen under usual assumptions (such as the convergence of F AnA * n toward1 a probability measure when N, n → ∞ and the convergence of the ratio N n -1 toward some positive constant c).

As will appear later, the reason of this lack of convergence lies in the fact that this additional term not only depends upon the spectrum of A n A * n but also on its eigenvectors. In order to avoid cumbersome assumptions enforcing the joint convergence of A n A * n 's spectrum and eigenvalues, we shall express our fluctuation results in the same way as in [START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF] and prove that the distribution of the linear statistics L n (f ) becomes close to a family of Gaussian distributions, whose parameters (mean and variance) may not converge. Namely, we shall establish that there exists a Gaussian random variable N (B n , Θ n ) such that :

d LP (L n (f ), N (B n , Θ n )) -----→ N,n→∞ 0 , (2.1.7)
where d LP denotes the L¨|vy-Prohorov distance (and in particular metrizes the convergence of laws).

Outline of the chapter

In Section 2.2, we state the main results of the chapter ; in Section 2.3, we establish the proof of the CLT (Theorem 2.2.1) except the identification of the covariance of the normalized trace, which is postponed to Section 2.4. In Section 2.5, we compute the limiting bias and prove Theorem 2.2.2.

Statement of the Central Limit Theorem

We state below the main assumptions of the chapter. Recall the fact that N = N (n) and the asymptotic regime (2.0.3) where N, n → ∞ and denote by

c n △ = N n , c - △ = lim inf N n and c + △ = lim sup N n . Assumption 2.2.1 The random variables (x n ij ; 1 ≤ i ≤ N (n), 1 ≤ j ≤ n, n ≥ 1)
are real or complex, independent and identically distributed (i.i.d.). They satisfy

Ex n ij = 0 , E|x n ij | 2 = 1 and E|x n ij | 4 < ∞ .
Associated to these moments are the quantities :

ϑ = E(x 2 ij ) and κ = E|x ij | 4 -|ϑ| 2 -2 .
Notice that in the case where the x ij 's are real, ϑ = 1 ; in the case where the random variables are complex with decorrelated real and imaginary part with equal variance, ϑ = 0.

Assumption 2.2.2 The family of deterministic N × n complex matrices (A n ) is bounded for the spectral norm : Denote by d LP the Lévy-Prohorov distance between two probability measures P, Q defined as :

a max △ = sup n≥1 A n < ∞. Assumption 2.2.3 Function f : R → R is analytic on an open region containing [0, ℓ + ] where ℓ + = 2(1 + √ c + ) 2 + 2a max . Remark 2.2.1 Since Y n Y * n ≤ 2(n -1 X n X * n +A n A * n ) ,
d LP (P, Q) = inf{ε > 0 : P (A) ≤ Q(A ε ) + ε for all Borel sets A ⊂ R d },
where A ε is an ε-blow up of A (see [18, Chapter 1, Section 6] for more details). If X and Y are random variables with distributions L(X) and L(Y ), we simply write (with a slight abuse of notations)

d LP (X, Y ) instead of d LP (L(X), L(Y )).
It is well-known that the Lévy-Prohorov distance metrizes the convergence in distribution (see for instance [START_REF] Dudley | Real analysis and probability[END_REF]Chapter 11]).

In the sequel, we shall often drop subscripts and superscripts n and for example write Q instead of Q n and x ij instead of x n ij , also we shall further simplify and denote :

δ 1,2 = δ(z 1,2 ) , and T 1,2 = T(z 1,2 ) ,
where δ and T are respectively defined in (2.1.2) and (2.1.4). We now introduce a number of quantities that will help in expressing the covariance and bias in the CLT.

γ(z 1 , z 2 ) = 1 n TrT 1 T 2 , γ(z 1 , z 2 ) = 1 n Tr T1 T2 , γ † (z 1 , z 2 ) = 1 n TrT 1 T T 2 , γ † (z 1 , z 2 ) = 1 n Tr T1 TT 2 , (2.2.1)
In order to express the variance and bias, we need the following notations : For M = I or T,

ν † M (z 1 , z 2 ) = 1 n TrA * T 1 MT T 2 Ā (1 + δ 1 )(1 + δ 2 ) , (2.2.2) ν † M (z 1 , z 2 ) = 1 n TrA T T 1 MT T 2 A (1 + δ 1 )(1 + δ 2 ) , (2.2.3) γ † M (z 1 , z 2 ) = 1 n TrT 1 MT T 2 (2.2.4) η(z 1 , z 2 ) = 1 n n j=1 z 1 z 2 tjj (z 1 ) tjj (z 2 ) (2.2.5) ω † (z 1 , z 2 ) = 1 n n k=1 n ℓ=1,ℓ =k a * k T 1 a ℓ a T ℓ T T 2 āk (1 + δ 1 ) 2 (1 + δ 2 ) 2 .
(2.2.6)

As we shall see, M will be equal to the identity matrix I in the computation of the covariance, and we will simply denote

ν † I = ν † , ν † I = ν † , γ † I = γ † .
In the computation of the bias where

z 1 = z 2 = z, M = T(z).
Remark 2.2.2 Notice that notation † refers to matrix products involving T and T T . In the case where matrix A is real, ν † M = ν † M ; however, these terms differ if A has complex entries.

Define :

∆ n (z 1 , z 2 ) = 1 - TrT 1 AA * T 2 n(1 + δ 1 )(1 + δ 2 ) 2 -z 1 z 2 γγ, (2.2.7)
∆ ϑ n (z 1 , z 2 ) = 1 -ϑν † 1 -θν † -|ϑ| 2 z 1 z 2 γ † γ † , (2.2.8)
Υ n (z) = 1 - z n TrT 2 - 1 n TrTAA * T (1 + δ) 2 -1 . (2.2.9) Remark 2.2.3 From (2.1.4),    T T (z) = -z 1 + δ(z) I N + Ā Ā * 1+δ(z) -1 TT (z) = -z (1 + δ(z)) I n + Ā * Ā 1+ δ(z) -1
.

Notice that T T (z) and TT (z) differ from T(z) and T(z) due to the presence of z. However, the following holds true :

T * (z) = T(z) = -z 1 + δ(z) I N + AA * 1 + δ(z) -1 .

Statement of the Central Limit Theorem

Recall that L n (f ) writes :

L n (f ) = - 1 2iπ Γ f (z)M n (z)dz = - 1 2iπ Γ f (z)M 1 n (z)dz - 1 2iπ Γ f (z)M 2 n (z)dz △ = L 1 n (f ) + L 2 n (f ) .
The first term above accounts for the fluctuations and is handled in Theorem 2.

△ = L 1 n (f 1 ), • • • , L 1 n (f k ) . Then d LP (L 1 n (f ), N 1 n (f )) -----→ N,n→∞ 0,
where

N 1 n (f ) is a centered Gaussian vector N k (0, V) with covariance matrix V = (V ij ; 1 ≤ i, j ≤ k) defined as V ij = - 1 4π 2 Γ 1 Γ 2 f i (z 1 )f j (z 2 )Θ n (z 1 , z 2 )dz 1 dz 2 , (2.2.10)
where

Θ n (z 1 , z 2 ) △ = Θ 0,n (z 1 , z 2 ) + Θ 1,n (ϑ, z 1 , z 2 ) + Θ 2,n (κ, z 1 , z 2 ) , (2.2.11) with Θ 0,n (z 1 , z 2 ) = s ′ n (z 1 )s ′ n (z 2 ) (s n (z 1 ) -s n (z 2 )) 2 - 1 (z 1 -z 2 ) 2 ,
(2.2.12)

Θ 1,n (ϑ, z 1 , z 2 ) = - ∂ ∂z 2 ∂∆ ϑ n ∂z 1 1 ∆ ϑ n , (2.2.13) Θ 2,n (κ, z 1 , z 2 ) = κ ∂ 2 ∂z 1 ∂z 2    z 1 z 2 n 2 N i=1 t ii (z 1 )t ii (z 2 ) n j=1 tjj (z 1 ) tjj (z 2 )    , (2.2

.14)

and where ∆ ϑ n is defined in (2.2.8) and where s n is defined as :

s n (z) = z(1 + δn )(1 + δ n ) , (2.2.15)
moreover, the contours Γ 1 , Γ 2 are non-overlapping, taken in the positive direction, and each enclosing [0, ℓ + ], where ℓ + is defined in Assumption 2.2.3.

In the case where the random variables x n ij 's and matrix A are real (in particular ϑ = 1), then Θ 0,n = Θ 1,n .

Proof of Theorem 2.2.1 closely follows the ideas in Bai and Silverstein [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF] and in Najim [START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF], and is postponed to Section 2.3, except for the computation of the covariance. This part of the proof is postponed to Section 2.4, and the computations therein are strongly inspired by those in Hachem et al. [START_REF] Hachem | A CLT for informationtheoretic statistics of non-centered Gram random matrices[END_REF].

(f ) -B(f ) -----→ N,n→∞ 0 , where B(f ) △ = - 1 2iπ Γ f (z)B(z) dz
with Γ a contour taken in the positive direction and enclosing [0, ℓ + ] (ℓ + being defined in Assumption 2.2.3) and

B(z) = Υ n ∆ ϑ n |ϑ| 2 γ † T γ † + ϑν † T + θν † T -|ϑ| 2 (ν † T ν † + ν † T ν † ) +|ϑ| 2 γ † T ω † + θ|ϑ| 2 η(ν † γ † T -ν † T γ † ) +κ z 2 Υ n n 2 n j=1 t2 jj N i=1 t ii [T 2 ] ii , (2.2

.16)

where all the relevant quantities are evaluated at (z 1 , z 2 ) = (z, z).

Specialization to the diagonal and Gaussian cases

Three cases deserve a specific statement : The (pseudo-)diagonal case, where a ij = 0 outside the diagonal, the real gaussian case and the complex gaussian case. Due to the orthogonal (resp. unitary) invariance of matrices with real (resp. complex) i.i.d. gaussian entries, these three cases are strongly connected, as we shall see.

The case where matrix A is pseudo-diagonal

When A is pseudo-diagonal, i.e. a ij = 0 for i = j, T is a diagonal matrix and T T = T. We notice that a * k Ta j = 0 when k = j, which implies that ω † = 0. Moreover, as all quantities depending on T and A are diagonal, we have :

ν † T = 1 n TrA * T 3 Ā (1 + δ) 2 , ν † T = 1 n TrA T T 3 A (1 + δ) 2 , γ † T = 1 n TrT 3 .
In this case, the bias writes :

B(z) = Υ n ∆ ϑ n z 2 |ϑ| 2 n 2 TrT 3 Tr T2 + ϑ n TrA * T 3 Ā (1 + δ) 2 + θ n TrA T T 3 A (1 + δ) 2 - |ϑ| 2 n 2 (1 + δ) 4 Tr(A * T 3 Ā)Tr(A T T 2 A) + Tr(A T T 3 A)Tr(A T T2 A) + θ|ϑ| 2 η n 2 (1 + δ) 2 Tr(A T T 3 A)Tr(T 2 ) -Tr(A T T 2 A)Tr(T 3 ) + κΥ n z 2 n 2 Tr T2 TrT 3 .
(2.2.17)

The case where matrix X is real gaussian and matrix A is real

In this case ϑ = 1 , κ = 0 , Ā = A , A * = A T and T T = T .
This yields minor simplifications in the bias formula (2.2.17). More important simplifications occur for the covariance since Θ 0,n = Θ 1,n and Θ 2,n = 0. Equation (2.2.11) writes

V ij = - 1 2π 2 f i (z 1 )f j (z 2 ) s ′ n (z 1 )s ′ n (z 2 ) (s n (z 1 ) -s n (z 2 )) 2 dz 1 dz 2 .
The case where matrix X is complex gaussian As ϑ = 0 and κ = 0, the bias is zero and the variance is

V ij = - 1 4π 2 f i (z 1 )f j (z 2 ) s ′ n (z 1 )s ′ n (z 2 ) (s n (z 1 ) -s n (z 2 )) 2 dz 1 dz 2 .
This is coherent with several papers. In complex gaussian case, it is well known that the bias is zero, (cf. [START_REF] Vallet | Improved subspace estimation for multivariate observations of high dimension : The deterministic signals case[END_REF][START_REF] Dumont | On the Capacity Achieving Covariance Matrix for Rician MIMO Channels : An Asymptotic Approach[END_REF]) while the variance has the same form as [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF].

Further computations

We further specialize the previous results and consider the case where A n = 0. A natural question is then to compare the results provided in Theorems 2.2.1 and 2.2.2 with those in Bai and Silverstein [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF] and in Najim [START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF], in the case where the considered population matrix is equal to the identity.

Comparison of the covariance

In the case where x ij are i.i.d. complex random variables with A = 0, κ = 0 and ϑ = 0, i.e.

Marčenko-Pastur case (cf. [START_REF] Marcenko | Distributions of eigenvalues for some sets of random matrices[END_REF]), we have δn (1

+ δ n ) = -1 z . Then s n (z) = - 1 + δn (z) δn (z) , and s ′ n (z 1 )s ′ n (z 2 ) (s n (z 1 ) -s n (z 2 )) 2 = δ′ n (z 1 ) δ′ n (z 2 ) δ2 n (z 1 ) δ2 n (z 2 ) 1+ δn(z1) δn(z1) -1+ δn(z2) δn(z2) 2 = δ′ n (z 1 ) δ′ n (z 2 ) ( δn (z 1 ) -δn (z 2 )) 2
which coincides with the formula in [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF].

Comparison of the bias

When A = 0, we will identify the bias (2.2.16) with the expression in Najim [START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF]Eq. (2.20), (2.21)] . Notice that in this case

T = -(z(1 + δ)) -1 I N , T = -(z(1 + δ)) -1 I n .
Taking trace and dividing by n, we have

δ = -c n (z(1 + δ)) -1 , δ = -(z(1 + δ)) -1 .
(2.2.18)

Injecting these terms into [70, Eq. (2.20), (2.21)], the bias is

B 1 (ϑ, z) = |ϑ| 2 c n δ3 (1 + δ) 3 1 -c n δ2 (1+ δ) 2 1 -c n |ϑ| 2 δ2 (1+ δ) 2 B 2 (κ, z) = κc n δ3 (1 + δ) 3 1 -c n δ2 (1+ δ) 2 .
With Theorem 2.2.2 , since for M = I or T,

ν † M = ν † M = 0, Υ n = 1 -c n 1 z(1 + δ) 2 -1 , ∆ ϑ n = 1 -c n |ϑ| 2 δ2 (1 + δ) 2 ,
we have

B 1 (ϑ, z) = - c n |ϑ| 2 δ2 z(1 + δ) 3 1 -c n 1 z(1+ δ) 2 1 -c n |ϑ| 2 δ2 (1+ δ) 2 B 2 (κ, z) = - κc n δ2 z(1 + δ) 3 1 -c n 1 z(1+ δ) 2 . To show B 1 (ϑ, z) = B 1 (ϑ, z), with comparison, it suffices to show that -z δ 1 -c n δ2 (1 + δ) 2 -1 = 1 -c n 1 z(1 + δ) 2 -1 , which is equivalent to show -z δ 1 -c n 1 z(1 + δ) 2 = 1 -c n δ2 (1 + δ) 2 .
From (2.2.18), we have

1 -c n δ2 (1 + δ) 2 + z δ 1 -c n 1 z(1 + δ) 2 = -zδ δ - c n δ 1 + δ = 0.
This proves B 1 (ϑ, z) = B 1 (ϑ, z). The same method yields also that B 2 (κ, z) = B 2 (κ, z).

2.3 Proof of Theorem 2.2.1 (I)

Outline of the proof

Recall that

L 1 n (f ) = - 1 2iπ Γ f (z) (TrQ n (z) -TrEQ n (z)) dz = - 1 2iπ Γ f (z)M 1 n (z) dz
where Γ is a positively oriented contour enclosing the limiting spectrum on which f is analytic. In order to establish the fluctuations of the term L 1 n (f ), we closely follow and adapt the strategy in Bai and Silverstein [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF]. We first establish the gaussian fluctuations of the process (M 1 n ) on the contour Γ and then prove that (M 1 n ) is tight over Γ. The gaussian fluctuations of M 1 n (z) are established via the following central limit theorem for martingales, which is a variation on [START_REF] Billingsley | Probability and Measure[END_REF]Theorem 35.12] (see also Lemmas 4.7 and 4.8 in [70]) :

Theorem 2.3.1 Suppose that for each n (Y nj ; 1 ≤ j ≤ r n ) is a C d -valued martingale difference
sequence with respect to the increasing σ-field {F n,j ; 1 ≤ j ≤ r n } having second moments. Write :

Y T nj = (Y 1 nj , • • • , Y d nj ) .
Assume moreover that (Θ n (k, ℓ)) n and ( Θn (k, ℓ)) n are uniformly bounded sequences of complex numbers, for

1 ≤ k, ℓ ≤ d. If rn j=1 E Y k nj Ȳ d nj | F n,j-1 -Θ n (k, ℓ) P ---→ n→∞ 0 , (2.3.1) rn j=1 E Y k nj Y ℓ nj | F n,j-1 -Θn (k, ℓ) P ---→ n→∞ 0 , (2.3.2)
and for each ε > 0, the following Lyapunov condition holds true :

rn j=1 E Y nj 2 I Y nj >ε ---→ n→∞ 0 . (2.3.3) Then d LP   rn j=1 Y nj , Z n   ---→ n→∞ 0 ,
where Z n is a C d -valued centered Gaussian random vector with parameters

EZ n Z * n = (Θ n (k, ℓ)) k,ℓ and EZ n Z T n = ( Θn (k, ℓ)) k,ℓ .
In view of Theorem 2.3.1, we shall study the fluctuations of

M 1 n (z 1 ), • • • , M 1 n (z d ) with (z 1 , • • • , z d ) ∈ C + .
Here are the different steps of the proof :

1. We first perform in Section 2.3.2 a double truncation step : we truncate the random variables (x ij )'s and modify the process (M n ) near the real axis where it may blow up.

The tightness of the process (M 1

n ) over a given contour is established in Section 2.3.5. 5. In order to establish the tightness of the gaussian process (N n (z), ; z ∈ Γ), we introduce a meta matrix model in Section 2.3.6.

Truncation

We closely follow Bai and Silverstein [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF] and proceed with two truncation steps. In the first step the entries are truncated at an appropriate level, then centered and normalized, with no effect on the fluctuations of the linear spectral statistics. Truncation of the entries essentially enables to deal with entries having a fourth moment finite, which is the optimal assumption for our results to hold. The second truncation step is concerned with the truncation of the process associated to the normalized resolvent near the real axis. Since

Tr Q n (z) = N i=1 1 λ i -z
may blow up on the real axis, it is replaced by a process that coincides with it except on a thin strip covering the real axis. If the strip is thin enough, there are no losses at a fluctuation level. This is packaged in the following results whose proofs are deferred to Appendix 2.6.1.

xij = xij -Ex ij E |x ij -Ex ij | 2 , xij = x ij I {|x ij |≤ηn √ n} and Ỹn = n -1/2 Xn + A n then Ex 2 ij ---→ n→∞ ϑ = Ex 2 ij , Ex 4 ij ---→ n→∞ E|x ij | 4 . Moreover Tr f (Y n Y * n ) -Tr f ( Ỹn Ỹ * n ) = o P (1)
as N, n → ∞.

We shall now introduce a truncated version of M n . Let µ ℓ < 0 and µ r > ℓ + where ℓ + is defined in Assumption 2.2.3 ; let d > 0 and let ε n be a real sequence decreasing to zero and satisfying, for some α ∈ (0, 1) :

ε n ≥ N -α .
Consider the contour

Γ = C ∪ C , (2.3.4)
where

C = {µ ℓ + iv; v ∈ [0, d]} ∪ {x + id; x ∈ [µ ℓ , µ r ]} ∪ {µ r + iv; v ∈ [0, d]}. (2.3.5)
Denote by

C n = {µ ℓ + iv; v ∈ [ε n /n, d]} ∪ {x + id; x ∈ [µ ℓ , µ r ]} ∪ {µ r + iv; v ∈ [ε n /n, d]}. (2.3.6)
We can now introduce the truncated process Mn (z) defined for z ∈ C by :

Mn (z) =          M n (z) for z ∈ C n M n (µ r + iε n /n) for x = µ r , v ∈ [0, ε n /n] M n (µ ℓ + iε n /n) for x = µ ℓ , v ∈ [0, ε n /n] ,
and for z ∈ C by Mn (z) = Mn (z).

f (z) M n (z) -Mn (z) dz = o P (1)
as N, n → ∞.

From now on, and without further mention below, we assume that the entries are truncated, centered and normalized (and write x ij instead of xij ), and that the process M n is truncated near the real axis (and write M n instead of Mn ).

More notations and useful estimates

Denote by y j , a j and r j the j-th columns of matrices Y n , A n and n -1/2 X n . Recall that

Q(z) = (Y n Y * n -zI N ) -1 . If z, z 1 , z 2 ∈ C + , let v = |Imz| , v i = |Imz i | , i = 1, 2 .
Denote by

Y j Y * j = YY * -y j y * j (2.3.7) Q j = (YY * -y j y * j -zI N ) -1 , (2.3.8) b j = 1 1 + n -1 ETrQ j + a * j (EQ j )a j , (2.3.9) b = 1 1 + n -1 ETrQ , (2.3.10)
γ j = y * j Q j y j -n -1 E(TrQ j ) -a * j (EQ j )a j , (2.3.11)
γj = y * j Q j y j -n -1 TrQ j -a * j Q j a j , (2.3
.12)

β j = 1 1 + y * j Q j y j , (2.3.13) βj = 1 1 + n -1 TrQ j + a * j Q j a j , (2.3.14)
α j = y * j Q 2 j y j -n -1 TrQ 2 j -a * j Q 2 j a j . ( 2

2.3.4

The Lyapunov condition for the process (M 1 n )

We first decompose M 1 n into martingale increments. Let E 0 denote the expectation and E j the conditional expectation with respect to the σ-field F n,j generated by {x ℓ , 1 ≤ ℓ ≤ j}. Noticing that

Q -Q j = -Q j y j y * j Q j β j , (2.3.16)
and

β j = βj -β j βj γj = βj -β2 j γj + β2 j β j γ2 j , (2.3.17)
we have

M 1 n = Tr(Q -EQ) = Tr n j=1 (E j -E j-1 )(Q -Q j ) = n j=1 (E j -E j-1 )Tr (Q -Q j ) Otherwise stated, (M 1 n (z 1 ), • • • M 1 n (z d )
) is a martingale difference sequence with respect to the increasing σ-field generated by X's columns and we are in position to apply Theorem 2.3.1. Write :

M 1 n = n j=1 (E j -E j-1 )Tr (Q -Q j ) = - n j=1 (E j -E j-1 )β j y * j Q 2 j y j = - n j=1 (E j -E j-1 )( βj -β2 j γj + β2 j β j γ2 j )(α j + n -1 TrQ 2 j + a * j Q 2 j a j ) (a) = - n j=1 E j ( βj α j -β2 j γj (n -1 TrQ 2 j + a * j Q 2 j a j )) + n j=1 (E j -E j-1 ) β2 j (γ j α j -β j α j γ2 j ) - n j=1 (E j -E j-1 ) β2 j β j γ2 j (n -1 TrQ 2 j + a * j Q 2 j a j ) , △ = n j=1 P j + n j=1 P ′ j + n j=1 P ′′ j (2.3.18)
where we have used in (a) the fact that conditionally on y j , the expectation of α j and γj is zero.

Consider (z ℓ , 1 ≤ ℓ ≤ d) in C + .
|P j (z ℓ )| 2 ≤ K < ∞ , (2.3.22) lim N,n n j=1 E d ℓ=1 |P j (z ℓ )| 2 I { d ℓ=1 |P j (z ℓ )| 2 ≥ε 2 } = 0 . (2.3.23) Let z ∈ C + and recall that v = |Imz|. Taking into account the fact that -z -1 βj (z), -z -1 β j (z)
and -z -1 b j (z) are Stieltjes transforms of probability measures, a fact that can be proved using [44, Proposition 2.2], we easily obtain the estimates :

max βj (z) z , β j (z) z , b j (z) z ≤ 1 v . ( 2

.3.24)

Write :

n j=1 E|(E j -E j-1 ) β2 j (z)γ j (z)α j (z)| 2 ≤ 4 n j=1 E| β2 j (z)γ j (z)α j (z)| 2 (a) ≤ |z| 4 v 4 n j=1 E|γ j (z)| 4 1/2 E|α j (z)| 4 1/2 (b) ≤ K η 4 n ,
where (a) follows from (2.3.24) and (b) from Lemma 2.3.3. A similar argument yields :

E n j=1 (E j -E j-1 )β j (z)α j (z)γ 2 j (z) 2 -----→ N,n→∞ 0.
Gathering the two previous results yields (2.3.20) ; (2.3.21) can be proved similarly. Recall that

P j (z) = -E j βj (z)α j (z) -β2 j (z)γ j (z)( 1 n TrQ 2 j (z) + a * j Q 2 j (z)a j ) . ( 2
E|P j (z)| 2 ≤ K |z| 2 v 2 E|α j (z)| 2 + |z| 4 v 8 E|γ(z)| 2 ≤ K n , (2.3
E|P j (z)| 4 ≤ K |z| 4 v 4 E|α j (z)| 4 + |z| 8 v 16 E|γ j (z)| 4 = o(n -1 ) . Now, n j=1 E d ℓ=1 |P j (z ℓ )| 2 I { d ℓ=1 |P j (z ℓ )| 2 ≥ε 2 } ≤ 1 ε 2 n j=1 E d ℓ=1 |P j (z ℓ )| 2 2 ≤ K ε 2 n j=1 d ℓ=1 E|P j (z ℓ )| 4 -----→ N,n→∞ 0
). Since M 1 n (z) = M 1 n (z), it suffices to prove n i=1 E M 1 n (z k )M 1 n (z ℓ ) | F n,i-1 -Θ n (z k , z ℓ ) P -----→ N,n→∞ 0
for z k , z ℓ ∈ Γ, with Θ n given by (2.2.11). Section 2.4 is devoted to the computation of this key-estimate. Therefore, we achieve the finite-dimensional convergence of the process M 1 n (z).

Tightness of the process (M 1 n )

We closely follow the computations of the tightness in Bai and Silverstein [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF]Section 3]. Based on [START_REF] Billingsley | Probability and Measure[END_REF]Theorem 13.1] after the truncation step (cf. Section 2.3.2), it suffices to prove that there exists z 0 ∈ C n and a constant K such that M 1 n (z) satisfies :

sup n E|M 1 n (z 0 )| 2 < ∞ and sup n,z 1 ,z 2 ∈Cn E|(M 1 n (z 1 ) -M 1 n (z 2 ))| 2 |z 1 -z 2 | 2 ≤ K. ( 2
YnY * n min < µ ℓ ) = 0 .
Proof : Since µ ℓ < 0, we clearly have P(λ

YnY * n min < µ ℓ ) = 0. Now P(λ YnY * n max > µ r ) ≤ P(2λ 1 n XnX * n max + 2λ AnA * n max > µ r ) = P(λ 1 n XnX * n max > µ r 2 -λ AnA * n max ) ≤ P(λ 1 n XnX * n max > (1 + √ c) 2 ) = o(n -k ) ,
where the last result follows from [11,

M j = Q k 1 j (z 1 )Q k 2 j (z 2 ) .
Then for z 1 , z 2 ∈ C n , q ∈ N, there exists a constant K p which depends only on p, such that

max sup z∈Cn E Q j (z) p , sup z∈Cn E|y * j Q q j (z)y j | p ≤ K qp , (2.3.29) E y * j M j y j -n -1 TrM j -a * j M j a j 2 ≤ K 2(k 1 +k 2 ) n , (2.3.30) E y * j M j y j -n -1 TrM j -a * j M j a j 4 ≤ K 4(k 1 +k 2 ) η 4 n n , (2.3.31) E|γ j (z)| 2 ≤ K 2 /n, E|γ j (z)| 4 ≤ K 4 η 4 n /n . ( 2
Q(z 1 ) -Q(z 2 ) = (z 1 -z 2 )Q(z 1 )Q(z 2 ) .
Since (E j -E j-1 )Q j (z 1 )Q j (z 2 ) = 0, we obtain :

M 1 n (z 1 ) -M 1 n (z 2 ) z 1 -z 2 = n j=1 (E j -E j-1 )TrQ(z 1 )Q(z 2 ) = n j=1 (E j -E j-1 )Tr (Q(z 1 )Q(z 2 ) -Q j (z 1 )Q j (z 2 )) .
Using (2.3.16), we have :

Q(z 1 )Q(z 2 ) -Q j (z 1 )Q j (z 2 ) = (Q(z 1 ) -Q j (z 1 ))(Q(z 2 ) -Q j (z 2 )) +(Q(z 1 ) -Q j (z 1 ))Q j (z 2 ) + Q j (z 1 )(Q(z 2 ) -Q j (z 2 )) = β j (z 1 )β j (z 2 )Q j (z 1 )y j y * j Q j (z 1 )Q j (z 2 )y j y * j Q j (z 2 ) -β j (z 1 )Q j (z 1 )y j y * j Q j (z 1 )Q j (z 2 ) -β j (z 2 )Q j (z 1 )Q j (z 2 )y j y * j Q j (z 2 ),
which yields

Tr(Q(z 1 )Q(z 2 ) -Q j (z 1 )Q j (z 2 )) = β j (z 1 )β j (z 2 )(y * j Q j (z 1 )Q j (z 2 )y j ) 2 -β j (z 1 )y * j Q 2 j (z 1 )Q j (z 2 )y j -β j (z 2 )y * j Q 2 j (z 2 )Q j (z 1 )y j .
Therefore,

M 1 n (z 1 ) -M 1 n (z 2 ) z 1 -z 2 = n j=1 (E j -E j-1 )β j (z 1 )β j (z 2 )(y * j Q j (z 1 )Q j y j ) 2 - n j=1 (E j -E j-1 )β j (z 1 )y * j Q 2 j (z 1 )Q j (z 2 )y j - n j=1 (E j -E j-1 )β j (z 2 )y * j Q 2 j (z 2 )Q j (z 1 )y j △ = T 1 + T 2 + T 3 . (2.3.34)
Our goal is to show the absolute second moment of (2.3.34) is uniformly bounded over the contour C n . From the relation β j (z) = b j (z)b j (z)β j (z)γ j (z), the term T 2 writes :

T 2 = n j=1 (E j -E j-1 )β j (z 1 )y * j Q 2 j (z 1 )Q j (z 2 )y j = n j=1 (E j -E j-1 ) b j (z 1 )y * j Q 2 j (z 1 )Q j (z 2 )y j -β j (z 1 )b j (z 1 )y * j Q 2 j (z 1 )Q j (z 2 )y j γ j (z 1 ) = n j=1 b j (z 1 )E j y * j Q 2 j (z 1 )Q j (z 2 )y j -n -1 TrQ 2 j (z 1 )Q j (z 2 ) -a * j Q 2 j (z 1 )Q j (z 2 )a j - n j=1 b j (z 1 )(E j -E j-1 )β j (z 1 )y * j Q 2 j (z 1 )Q j (z 2 )y j γ j (z 1 ) △ = W 1 -W 2 .
By Lemma 2.3.5 with k 1 = 2, k 2 = 1 and (2.3.33), we have

E|W 1 | 2 = n j=1 |b j (z)| 2 E E j [y * j Q 2 j (z 1 )Q j (z 2 )y j -n -1 TrQ 2 j (z 1 )Q j (z 2 ) -a * j Q 2 j (z 1 )Q j (z 2 )a j ] 2 ≤ K. (2.3.35)
For W 2 , taking η ℓ ∈ (µ ℓ , 0) and η r ∈ (ℓ + , µ r ) we have :

E|W 2 | 2 = n j=1 |b j (z 1 )| 2 E|(E j -E j-1 )β j (z 1 )y * j Q 2 j (z 1 )Q j (z 2 )y j γ j (z 1 )| 2 (a) ≤ K n j=1 E β j (z 1 )y * j Q 2 j (z 1 )Q j (z 2 )y j γ j (z 1 ) 2 (b) ≤ K n j=1 E|γ j (z 1 )| 2 + v -8 P(λ Y j Y * j max ≥ η r or λ Y j Y * j min ≤ η ℓ ) (c) ≤ K,
where (a) uses (2.3.33), (b) is from the fact that β j (z 1 )y * j Q 2 j (z 1 )Q j (z 2 )y j is bounded when (2.3.30). The term T 3 in (2.3.34) can be handled similarly. We now handle T 1 :

η ℓ ≤ λ Y j Y * j max ≤ η r and (c) is from
n j=1 (E j -E j-1 )β j (z 1 )β j (z 2 )(y * j Q j (z 1 )Q j (z 2 )y j ) 2 = n j=1 b j (z 1 )b j (z 2 )(E j -E j-1 )[(y * j Q j (z 1 )Q j (z 2 )y j ) 2 -(n -1 TrQ j (z 1 )Q j (z 2 ) + a * j Q j (z 1 )Q j (z 2 )a j ) 2 ] - n j=1 b j (z 2 )(E j -E j-1 )β j (z 1 )β j (z 2 )(y * j Q j (z 1 )Q j y j ) 2 γ j (z 2 ) - n j=1 b j (z 1 )b j (z 2 )(E j -E j-1 )β j (z 1 )(y * j Q j (z 1 )Q j y j ) 2 γ j (z 2 ) △ =Y 1 -Y 2 -Y 3 .
Both Y 2 and Y 3 can be handled in the same way as W 2 and one can prove that :

E|Y 2 | 2 ≤ K, E|Y 3 | 2 ≤ K.
Moreover,

E|Y 1 | 2 ≤ K n j=1 E|(y * j Q j (z 1 )Q j (z 2 )y j ) 2 -(n -1 TrQ j (z 1 )Q j (z 2 ) + a * j Q j (z 1 )Q j (z 2 )a j ) 2 | 2 (a) ≤ K n j=1 E|y * j Q j (z 1 )Q j (z 2 )y j -n -1 TrQ j (z 1 )Q j (z 2 ) -a * j Q j (z 1 )Q j (z 2 )a j | 2 + Kv -4 1 v -4 2 P(λ Y j Y * j max ≥ η r or λ Y j Y * j min ≤ η ℓ ) (b) ≤ K
where (a) comes from the fact that

Q j (z 1 )Q j (z 2 ) is bounded when η ℓ ≤ λ Y j Y * j
max ≤ η r and (b) is again a consequence of (2.3.30) and Lemma 2.3.4.

We conclude that all terms in (2.3.34) are indeed bounded uniformly over C n . This completes the proof of the tightness of the process M 1 n (z).

Tightness of the Gaussian process

The last task consists in establishing the tightness for the Gaussian process N n (z) = N (0, Θ n (z)) where Θ n is defined in (2.2.11). We will transfer the tightness of the process M 1 n (z) to the tightness of the Gaussian process N n (z) with the help of meta-model.

With the extra parameter M , consider the N M × nM matrix

A M n =     A n • • • 0 . . . . . . . . . 0 • • • A n    
where A M n is the block matrix whose diagonal is composed of M matrices A n and 0 elsewhere. Consider now the random matrix

Y M n = 1 √ nM X M n + A M n . Denote M 1 n,M (z) = 1 N M N M j=1 1 λ j -z -1 N M N M j=1 E 1 λ j -z where (λ i ) 1≤i≤N M are eigenvalues of Y M n Y M n * .
Cov[M 1 n,M (z 1 ), M 1 n,M (z 2 )] ----→ M →∞ Θ n (z 1 , z 2 )
where

Θ(z 1 , z2) = Θ 0,n (z 1 , z 2 ) + Θ 1,n (ϑ, z 1 , z 2 ) + Θ 2,n (κ, z 1 , z 2 ).
With these notations at hand, we shall show the tightness of the Gaussian process. According to [18, Chapter 2, Section 7], a sequence of Gaussian process is tight if and only if it is relatively compact in distribution. Consider the set of matrices :

{A M n : M ≥ 1, A n is an N × n matrix

Proof of Theorem 2.2.1 (II) : Computation of the covariance

In Hachem et al. [START_REF] Hachem | A CLT for informationtheoretic statistics of non-centered Gram random matrices[END_REF], a CLT for mutual information has been established

1 N log det(Y n Y * n + ρI), for ρ > 0.
Be aware that

Tr(Y n Y * n + ρI) -1 = ∂ ∂ρ log det(Y n Y * n + ρI),
and based on this idea, in this section we shall calculate the covariance Cov(M 1 n (z 1 ), M 1 n (z 2 )). We denote :

Γ j (z) = y * j Q j (z)y j -( 1 n TrQ j (z) + a * j Q j (z)a j ) 1 + 1 n TrQ j (z) + a * j Q j (z)a j , (2.4.1) bj (z) = -1 z(1 + a * j Q j (z)a j + 1 n TrQ j (z)) , (2.4.2) qjj (z) = -1 z(1 + y * j Q j (z)y j ) , (2.4.3)
T j (z) = -z(1 + δn )I N + A j A * j 1 + δ n (z) -1 , (2.4.4) tjj = -1 z(1 + a * j T j a j + δ n ) , ( 2

.4.5)

where A j is the matrix A where the j-th column has been removed and δ n , δn are defined in (3.1.2). From (2.4.1)-(2.4.5), we have then

Q(z) = Q j (z) + z qjj (z)Q j (z)y j y * j Q j (z), (2.4.6) Q j (z) = Q(z) + Q(z)y j y * j Q(z) 1 -y * j Q(z)y j , (2.4.7
)

1 + y * j Q j (z)y j = 1 1 -y * j Q(z)y j , (2.4.8)
y * j Q(z) = -z qjj (z)y * j Q j (z), (2.4
.9)

1 + z tℓℓ a * ℓ T ℓ a ℓ = -z tℓℓ (1 + δ), (2.4.10) -z tjj a * j T j b = a * j Tb 1 + δ n
, for any vector b, (2.4.11) which will be extremely useful in the sequel.

The following theorem gives the estimates of the terms.

Theorem 2.4.1 [39, Theorem 3.1] Let (u n ) and (v n ) be two sequences of deterministic complex

N × 1 vectors bounded by sup n≥1 max( u n , v n ) < ∞,
and let (U n ) be a sequence of deterministic N × N matrix with bounded spectral norm

sup n≥1 U n < ∞.
Then, we have the following inequalities,

n j=1 E|u * n Q j a j | 2 ≤ K, ( 2
.4.12)

1 n TrU(T -EQ) ≤ K n , (2.4.13)
E|u * n (Q -T)v n | 2 ≤ K p n , (2.4.14)
E|u * n (Q j -T j )v n | 2 ≤ K p n , (2.4.15
)

E|TrU(Q -EQ)| 2 ≤ K, ( 2
A n (z 1 , z 2 ) = n j=1 E j-1 {E j Γ j (z 1 )E j Γ j (z 2 )} .
(2.4.20)

Then ∂ 2 ∂z 1 ∂z 2 A n (z 1 , z 2 ) = (2.4.19), (2.4.21)
and if there exists a term L n (z 1 , z 2 ) such that

d LP (A n (z 1 , z 2 ), L n (z 1 , z 2 )) -----→ N,n→∞ 0, then d LP Cov(M 1 n (z 1 ), M 1 n (z 2 )), ∂ 2 ∂z 1 ∂z 2 L n (z 1 , z 2 ) -----→ N,n→∞ 0
Cov(M 1 n (z 1 ), M 1 n (z 2 )) = Θ 0,n (z 1 , z 2 ) + Θ 1,n (ϑ, z 1 , z 2 ) + Θ 2,n (κ, z 1 , z 2 ).
(2.4.23)

Remark 2.4.1 In [START_REF] Hachem | A CLT for informationtheoretic statistics of non-centered Gram random matrices[END_REF], they have shown that for z = -ρ with ρ > 0,

Var(A n (z)) = -log ∆ n (z, z) -log ∆ ϑ n (z, z) + κz 2 n 2 N i=1 t 2 ii (z) n j=1 t2 jj (z) + o(1).
The main task is to generate the precedent formula of Cov(A n (z 1 ), A n (z 2 )) for z 1 , z 2 ∈ C + . However, log z 1 z 2 = log z 1 + log z 2 is no more true in complex case. We follow closely the method in [START_REF] Hachem | A CLT for informationtheoretic statistics of non-centered Gram random matrices[END_REF] and will conquer this difficulty. In addition, [START_REF] Hachem | A CLT for informationtheoretic statistics of non-centered Gram random matrices[END_REF] supposes that E|x ij | 16 < ∞ which will be relaxed later. In particular, the moment condition will be replaced by Lemma 2.3.3 in the proof of Lemma 2.4.6.

Proof : We begin with the proof of (2.4.21) in Proposition 2.4.1. By Cauchy-Schwarz,

|E(M 1 n (z 1 )M 1 n (z 2 ))| ≤ (E|M 1 n (z 1 )| 2 ) 1/2 (E|M 1 n (z 2 )| 2 ) 1/2 .
With the dominated convergence theorem with expectation, it suffices to show that sup n E|M n (z 1 )|

n j=1 E j-1 {E j Γ j (z 1 )E j Γ j (z 2 )} ≤ K|z 1 z 2 |(v 1 v 2 ) -2 .
And by the argument of meta-model (cf. Section 2.3.6), the above inequality implies that n j=1 E j-1 {E j Γ j (z 1 )E j Γ j (z 2 )} is bounded. Then the limit L n is in the closure of A n , hence also bounded. This implies that L n is bounded. Now we will apply Lemma 2.4.1. Denote v 0 a lower bound of |Imz i | and {z i } ⊂ D = {z : v 0 ≤ |Imz| ≤ K}. Then by a diagonalization argument, for any subsequence of the natural numbers, there is a further subsequence such that, with probability one, A n -L n converges for each pair z k , z ℓ ∈ {z i }. Write A n -L n as f n (z 1 , z 2 ). We concentrate on this subsequence and on one realization for which convergence holds. For each z ℓ ∈ {z i }, we apply Lemma 2.4.1 on the set of {z : v 0 /2 ≤ Imz ≤ K} ∪ {z : -K ≤ Imz ≤ -v 0 /2} to get the convergence of f n (z, z ℓ ) to 0, satisfying ∂ ∂z f n (z, z l ) → 0. From Lemma 2.4.1 we see that ∂ ∂z f n (z, w) is bounded in w and n for all w ∈ D. Applying again Lemma 2.4.1 on the remaining variable we see that ∂ 2 ∂z∂w f n (z, z l ) → 0. This proves (2.4.22).

Thus, it suffices to concentrate on the convergence of A n which is the central object in the next section.

Computation of the covariance of A n (z)

In this section , we will calculate the limit of A n (z) defined in (2.4.20). Noticing Γ j = -z bj γj , we first prove that

n j=1 E j-1 (E j z 1 bj (z 1 )γ j (z 1 ))(E j z 2 bj (z 2 )γ j (z 2 )) - n j=1 z 1 z 2 tjj (z 1 ) tjj (z 2 )E j-1 (E j γj (z 1 ))(E j γj (z 2 )) P ---→ n→∞ 0. ( 2
E E j-1 {E j [z 1 bj (z 1 )γ j (z 1 )]E j [z 2 bj (z 2 )γ j (z 2 )]} -z 1 z 2 tjj (z 1 ) tjj (z 2 )E j-1 {E j γj (z 1 )E j γj (z 2 )} ≤ E z 1 z 2 E j-1 {E j [( bj (z 1 ) -tjj (z 1 ))γ j (z 1 )]E j [ bj (z 2 )γ j (z 2 )]} -z 1 z 2 tjj (z 1 )E j-1 {E j [γ j (z 1 )E j ( bjj (z 2 ) -tjj (z 2 ))γ j (z 2 )]} ≤ KE 1/2 | bj (z 1 ) -tjj (z 1 )| 2 E 1/2 |γ j (z 1 )E j [γ j (z 2 )]| 2 +KE 1/2 | bj (z 2 ) -tjj (z 2 )| 2 E 1/2 |γ j (z 2 )E j [γ j (z 1 )]| 2 ≤ Kη 2 n n .
By summing the precedent inequalities over j, we achieve Equation (2.4.24). Now we will calculate the covariance. The formula below plays a key role in the calculation.

Proposition 2.4.2 [39, Equation (3.20)] Let x = (x 1 , • • • , x N )
T be an N × 1 vector where x i are centered i.i.d. complex random variables with unit variance. Let y = N -1/2 x, M = (m ij ) and P = (p ij ) be N × N deterministic complex matrices and let u be an N × 1 deterministic vector. vdiag(M) stands for the

N × 1 vector [M 11 , • • • , M N N ] T .
Denote Υ(M), the random variable :

Υ(M) = (y + u) * M(y + u).
Then EΥ(M) = 1 N TrM + u * Mu and the covariance between Υ(M) and Υ(P) is :

E[(Υ(M) -EΥ(M))(Υ(P) -EΥ(P))] = 1 N 2 Tr(MP) + 1 N (u * MPu + u * PMu) + |E[x 1 ] 2 | 2 N 2 Tr(MP T ) + E[x 2 1 ] N u * PM T ū + E[x 2 1 ] N u T M T Pu + E[|x 1 | 2 x 1 ] N 3/2 (u * Pvdiag(M) + u * Mvdiag(P)) + E[|x 1 | 2 x1 ] N 3/2 (vdiag(P) T Mu + vdiag(M) T Pu) + κ N 2 N i=1 m ii p ii , where κ = E|x 1 | 4 -2 -|Ex 2 1 | 2 .
Noticing that E j (Q j (z 1 )) is F n,j-1 measurable, by the precedent formula with M = E j (Q j (z 1 )),

P = E j (Q j (z 2 )), we have n j=1 z 1 z 2 tjj (z 1 ) tjj (z 2 )E j-1 [E j γj (z 1 )E j γj (z 2 )] △ = n j=1 ξ 1j + n j=1 ξ 2j + n j=1 ξ ′ 2j + n j=1 ξ 3j + n j=1 ξ 4j ,
where

n j=1 ξ 1j △ = κ n 2 n j=1 z 1 z 2 tjj (z 1 ) tjj (z 2 ) N i=1 E j [Q j (z 1 )] ii E j [Q j (z 2 )] ii (2.4.25) n j=1 ξ 2j △ = 1 n n j=1 z 1 z 2 tjj (z 1 ) tjj (z 2 )E(|x 11 | 2 x 11 ) × (2.4.26) a * j (E j Q j (z 1 ))vdiag(E j Q j (z 2 )) √ n + a * j (E j Q j (z 2 ))vdiag(E j Q j (z 1 )) √ n n j=1 ξ ′ 2j △ = 1 n n j=1 z 1 z 2 tjj (z 1 ) tjj (z 2 )E(|x 11 | 2 x11 ) × (2.4.27) vdiag(E j Q j (z 2 )) T (E j Q j (z 1 ))a j √ n + vdiag(E j Q j (z 1 )) T (E j Q j (z 2 ))a j √ n n j=1 ξ 3j △ = 1 n n j=1 z 1 z 2 tjj (z 1 ) tjj (z 2 ) 1 n Tr(E j Q j (z 1 ))(E j Q j (z 2 )) (2.4.28) +a * j (E j Q j (z 1 ))(E j Q j (z 2 ))a j + a * j (E j Q j (z 2 ))(E j Q j (z 1 ))a j n j=1 ξ 4j △ = 1 n n j=1 z 1 z 2 tjj (z 1 ) tjj (z 2 ) |ϑ| 2 n Tr(E j Q j (z 1 ))(E j Q T j (z 2 )) (2.4.29) +ϑa * j (E j Q j (z 2 ))(E j Q T j (z 1 ))ā j + θa T j (E j Q T j (z 1 ))(E j Q j (z 2 ))a j .
n j=1 ξ 1j contributes the term with κ in (2.4.23). n j=1 ξ 2j and n j=1 ξ ′ 2j will not contribute in the final expression. n j=1 ξ 3j corresponds to the term Θ 0,n while n j=1 ξ 4j corresponds to ∆ ϑ n . The first two terms are easy to compute. The computations for n j=1 ξ 3j and n j=1 ξ 4j are similar which demand more carefulness and a good understanding how the terms interact. We start the computations with n j=1 ξ 1j .

Lemma 2.4.2 Recall the expression of n j=1 ξ 1j in (2.4.25). We have

n j=1 ξ 1j - κz 1 z 2 n 2 N i=1 t ii (z 1 )t ii (z 2 ) n j=1 tjj (z 1 ) tjj (z 2 ) P -----→ N,n→∞ 0. Proof : The idea is trying to replace E j [Q j ] ii by t ii . Write 1 n N i=1 E j [Q j (z 1 )] ii E j [Q j (z 2 )] ii - 1 n N i=1 E j [Q j (z 1 )] ii t ii (z 2 ) = 1 n N i=1 E j [Q j (z 1 )] ii E j ([Q j (z 2 )] ii -[Q(z 2 )] ii ) + 1 n N i=1 E j [Q j (z 1 )] ii (E j [Q(z 2 )] ii -t ii (z 2 )) △ =ε 1,j + ε 2,j .
To estimate ε 1,j , we use rank-one perturbation lemma (see [START_REF] Hachem | A CLT for Information Theoretic Statistics of Gram Random Matrices with a Given Variance Profile[END_REF]Lemma 6.3] and [83, Lemma 2.6]).

Lemma 2.4.3 The resolvent Q and the perturbed resolvent Q j satisfy for z = 0 and any

N × N matrix A : |TrA(Q -Q j )| ≤ A |Im(z)| .
Thanks to the rank-one perturbation lemma, the first term

ε 1,j = 1 n |E j Tr[diag(E j Q j (z 1 ))(Q j (z 2 ) -Q(z 2 ))]|
is with the order O(n -1 ). The second item

E|ε 2,j | = O(n -1/2 )
is a direct consequence of Cauchy-Schwarz and (2.4.18) in Lemma 2.4.1. Hence

n j=1 ξ 1j - κz 1 z 2 n 2 N i=1 n j=1 tjj (z 1 ) tjj (z 2 )E j [Q j (z 1 )] ii t ii (z 2 ) P ---→ n→∞ 0.
With the same method, we can replace E j [Q j ] ii (z 1 ) by t ii (z 1 ) successively without losing the order. This leads to the result. Proof : n j=1 ξ 2j and n j=1 ξ ′ 2j can be treated similarly, we focus on the proof of n j=1 ξ 2j . From

E n j=1 ξ 2j ≤ K n n j=1 E a * j (E j Q j (z 1 ))vdiag(Q j (z 2 )) √ n + K n n j=1 E a * j (E j Q j (z 2 ))vdiag(Q j (z 1 )) √ n ,
it suffices to consider the first term. The other term can be treated in the same way. Consider the term

1 n n j=1 E a * j (E j Q j (z 1 ))vdiag(Q j (z 2 )) √ n ≤ 1 n n j=1 E a * j Q j (z 1 )vdiag(T(z 2 )) √ n + 1 n n j=1 E a * j (E j Q j (z 1 ))vdiag(Q(z 2 ) -T(z 2 )) √ n + 1 n n j=1 E a * j (E j Q j (z 1 ))vdiag(Q j (z 2 ) -Q(z 2 )) √ n . (2.4.30)
The first term in (2.4.30) satisfies

n j=1 E a * j Q j (z 1 )vdiag(T(z 2 )) √ n ≤ √ n   n j=1 E a * j Q j (z 1 )vdiag(T(z 2 )) √ n 2   1/2 . As n -1/2 vdiag(T) = (n -1 n i=1 t 2 ii ) 1/2 ≤ K, (2.4.
12) in Theorem 2.4.1 can be applied and

1 n n j=1 E a * j Q j (z 1 )vdiag(T(z 2 )) √ n = O(n -1/2 ).
We now deal with the second term in (2.4.30) :

1 n n j=1 E a * j (E j Q j (z 1 ))vdiag(Q(z 2 ) -T(z 2 )) √ n ≤ KE vdiag(Q(z 2 ) -T(z 2 )) √ n ≤ K 1 n N i=1 E|q ii (z 2 ) -t ii (z 2 )| 2 1/2 = O(n -1/2 ),
by Equation (2.4.18) in Theorem 2.4.1. Since a * j (E j Q j ) is uniformly bounded by K, the third term in (2.4.30) can be estimated by using Lemma 2.4.3 :

1 n n j=1 E a * j (E j Q j (z 1 ))vdiag(Q j (z 2 ) -Q(z 2 )) √ n = 1 n 3/2 n j=1 E|Tr[diag(a * j (E j Q j (z 1 )))(Q j (z 2 ) -Q(z 2 ))]| ≤ K √ n .
Now we concentrate on the proof of the following lemma whose idea is from [START_REF] Hachem | A CLT for informationtheoretic statistics of non-centered Gram random matrices[END_REF]Lemma 4.3].

Lemma 2.4.5 Recall the expression of n j=1 ξ 3j in (2.4.28). Under the same assumptions, we have

∂ 2 ∂z 1 ∂z 2 n j=1 ξ 3j - s ′ n (z 1 )s ′ n (z 2 ) (s n (z 1 ) -s n (z 2 )) 2 + 1 (z 1 -z 2 ) 2 P ---→ n→∞ 0, where s n (z) = z(1 + δ)(1 + δ).
Proof : For future use, for a Hermitian matrix M with bounded spectral norm, consider the notations :

ψ M j (z 1 , z 2 ) = 1 n TrE {[E j Q(z 1 )]M[E j Q(z 2 )]} = 1 n TrE{[E j Q(z 1 )]MQ(z 2 )}, ζ M kj (z 1 , z 2 ) = E{a * k [E j Q(z 1 )]M[E j Q(z 2 )]a k } = E{a * k [E j Q(z 1 )]MQ(z 2 )a k }, θ M kj (z 1 , z 2 ) = E{a * k [E j Q k (z 1 )]M[E j Q k (z 2 )a k ]} = E{a * k [E j Q k (z 1 )]MQ k (z 2 )a k }, φ M j (z 1 , z 2 ) = 1 n j k=1 z 1 z 2 tkk (z 1 ) tkk (z 2 )θ M kj .
Remark 2.4.3 In particular, we will take M = I N to calculate the variance and M = T in the calculation of the bias.

We will decompose n j=1 ξ 3j . The following lemma allows to replace n j=1 ξ 3j by its expectation.

Lemma 2.4.6 For any N × 1 vector a with bounded Euclidean norm, we have

max j Var[a * E j Q(z 1 )ME j Q(z 2 )a] = o(n -1/2 ) and max j Var[TrE j Q(z 1 )ME j Q(z 2 )] = o(n 1/2 ).
The proof is postponed to Appendix 2.6.6.

Remark 2.4.4 In [39, Lemma 5.1], with the hypothesis E|x ij | 16 < ∞, it has been shown that

max j Var[a * E j Q(z 1 )ME j Q(z 2 )a] = o(n -1/2 ) = O(n -1
) which is stronger than the precedent bound. However, Lemma 2.4.6 is sufficient to ensure the replacement of n j=1 ξ 3j by its expectation.

Remark 2.4.5 Although the computation is a little lengthy, there is a key rule during the computation. Let y j = r j + a j . For matrices M 1 , M 2 with bounded spectral norms and independent of y j . Consider the term

Er * j M 1 y j y * j M 2 a j .
Developing y j y j = r j r * j + a j r * j + r j a * j + a j a * j , we will show that the only term which is not negligible is r j a * j , then we have :

Er * j M 1 y j y * j M 2 a j = Er * j M 1 r j a * j M 2 a j + o(1).
By Lemma 2.4.6, r * j M 1 r j and a * j M 2 a j are uncorrelated, and we get

Er * j M 1 y j y * j M 2 a j = 1 n ETrM 1 Ea * j M 2 a j + o(1).
The rest of the computation is the use of identities. We will be more precise during the proof.

By Lemma 2.4.6 and 2.4.3, observe that

n i=1 ξ 3j = 1 n n j=1 z 1 z 2 tjj (z 1 ) tjj (z 2 ) 1 n TrE[E j Q(z 1 )E j Q(z 2 )] +θ I N jj (z 1 , z 2 ) + θ I N jj (z 2 , z 1 ) + o(n -1/2 ),
Taking M = I N , it suffices to study

1 n n j=1 z 1 z 2 tjj (z 1 ) tjj (z 2 ) ψ I N j + θ I N jj (z 1 , z 2 ) + θ I N jj (z 2 , z 1 ) .
To simplify the notations, all terms which depend on z 1 (resp. z 2 ) will be denoted by

Q 1 = Q(z 1 ), T 1 = T(z 1 ), Q 1,ℓ = Q ℓ (z 1 ), (resp. Q 2 = Q(z 2 ), T 2 = T(z 2 ),Q 2,ℓ = Q ℓ (z 2 )
) etc. We will look for the equations verified by ζ M kj , ψ M j and φ M j . The idea is to have the terms which are displayed in Remark 2.4.5. There are four steps. In the first step, starting with ζ M kj , with

A -1 -B -1 = -A -1 (A -B)B -1
, we obtain the first relation satisfied by ζ M kj , ψ M j and φ M j . In the second step, beginning with ψ j , we search for the relation between ψ M j and φ M j . In Step 3, we start again with ζ M kj , with (2.4.6), we achieve the third equation with ζ M kj , ψ M j and φ M j . Hence, we obtain three equations with three parameters. This allows us to conclude the computation which is the goal in Step 4.

Step 1 : Expression of ζ M kj In this section, we will show that :

ζ M kj , ψ M j , φ M j satisfy ζ M kj = a * k T 1 MT 2 a k + ψ M j j ℓ=1 a * k T 1 a ℓ a * ℓ T 2 a k (1 + δ 1 )(1 + δ 2 ) + a * k T 1 T 2 a k 1 n j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ + a * k T 1 T 2 a k φ M j + O(n -1/2 ).
(2.4.31)

Writing

Q = T + T(T -1 -Q -1 )Q = T + T(-z δI N + (1 + δ) -1 AA * -YY * ) -1 Q, (2.4.32)
we have

ζ M kj = E[a * k (E j Q 1 )M(E j Q 2 )a k ] = E[a * k (E j [T 1 + T 1 (-z 1 δ1 I N + (1 + δ 1 ) -1 AA * -YY * )Q 1 ]MQ 2 a k ] = E[a * k T 1 MQ 2 a k ] -z 1 δ1 E[a * k T 1 (E j Q 1 )MQ 2 a k ] + (1 + δ 1 ) -1 E[a * k T 1 AA * (E j Q 1 )MQ 2 a k ] -E[a * k T 1 (E j YY * Q 1 )MQ 2 a k ] = a * k T 1 MT 2 a k -z 1 δ1 E[a * k T 1 (E j Q 1 )MQ 2 a k ] + X + Z + ε, (2.4.33)
where

X △ = (1 + δ 1 ) -1 E[a * k T 1 AA * (E j Q 1 )MQ 2 a k ], (2.4.34)
Z △ = -E[a * k T 1 (E j YY * Q 1 )MQ 2 a k ], (2.4.35)
and |ε| = O(n -1/2 ) by (2.4.14) in Theorem 2.4.1. Recall y j = r j + a j , the term X is

X = (1 + δ 1 ) -1 n ℓ=1 E[a * k T 1 a ℓ a * ℓ (E j Q 1 )MQ 2 a k ] = (1 + δ 1 ) -1 n ℓ=1 E[a * k T 1 a ℓ a * ℓ (E j Q 1,ℓ )MQ 2 a k ] +(1 + δ 1 ) -1 n ℓ=1 E[z 1 t1,ℓℓ a * k T 1 a ℓ a * ℓ (E j Q 1,ℓ y ℓ y * ℓ Q 1,ℓ )MQ 2 a k ] + ǫ 1 = (1 + δ 1 ) -1 n ℓ=1 E[a * k T 1 a ℓ a * ℓ (E j Q 1,ℓ )MQ 2 a k ] +(1 + δ 1 ) -1 n ℓ=1 E[z 1 t1,ℓℓ a * k T 1 a ℓ a * ℓ (E j Q 1,l a ℓ y * ℓ Q 1,ℓ )MQ 2 a k ] + ǫ 1 + ǫ 2 = (1 + δ 1 ) -1 n ℓ=1 E[a * k T 1 a ℓ a * ℓ (E j Q 1,l )MQ 2 a k ] +(1 + δ 1 ) -1 n ℓ=1 E[z 1 t1,ℓℓ a * k T 1 a ℓ a * ℓ T 1,ℓ a ℓ (E j y * ℓ Q 1,ℓ )MQ 2 a k ] + ǫ 1 + ǫ 2 + ǫ 3 △ = X 1 + X 2 + ǫ 1 + ǫ 2 + ǫ 3 , (2.4.36)
where

ǫ 1 △ = (1 + δ 1 ) -1 n ℓ=1 E[ t1,ℓℓ a * k T 1 a ℓ (E j (z 1 q1,ℓℓ -z 1 t1,ℓℓ )a * ℓ Q 1,ℓ y ℓ y * ℓ Q 1,ℓ )MQ 2 a k ], ǫ 2 △ = (1 + δ 1 ) -1 n ℓ=1 E[z 1 t1,ℓℓ a * k T 1 a ℓ a * ℓ (E j Q 1,ℓ r ℓ y * ℓ Q 1,ℓ )MQ 2 a k ], ǫ 3 △ = (1 + δ 1 ) -1 n ℓ=1 E[z 1 t1,ℓℓ a * k T 1 a ℓ (E j a * ℓ (Q 1,ℓ -T 1,ℓ )a ℓ y * ℓ Q 1,ℓ )MQ 2 a k ].
Using Formula (2.4.3) and (2.4.9),

ǫ 1 = (1 + δ 1 ) -1 E[E j (a * k T 1 Adiag(χ 1,ℓ )Y * Q 1 )MQ 2 a k ], (2.4.37)
where

χ 1,ℓ = -z 1 (q 1,ℓℓ -t1,ℓℓ )(1 + y * ℓ Q 1,ℓ y ℓ )a * ℓ Q 1,ℓ y ℓ . Notice that Y * Q ≤ Y Q < ∞,
where by truncation, Y < ∞ and Q ≤ |Imz| -1 . The same method yields that Y * Q j is also bounded and we obtain

|ǫ 1 | ≤ KE a * k T 1 Adiag(χ 1,ℓ ) ≤ KE 1/2 n ℓ=1 |[a * k T 1 A] ℓ | 2 |χ 1,ℓ | 2 ≤ KE 1/2 n ℓ=1 |[a * k T 1 A] ℓ | 2 |q 1,ℓℓ -t1,ℓℓ | 2 |a * ℓ Q 1,ℓ y ℓ | 2 ≤ KE 1/2 n ℓ=1 |[a * k T 1 A] ℓ | 2 |q 1,ℓℓ -t1,ℓℓ | 2 ≤ K √ n ,
where the last inequality follows from (2.4.18) in Theorem 2.4.1. A similar result for ǫ 2 and ǫ 3 can be shown that ǫ 2 = O(n -1/2 ) and ǫ 3 = O(n -1/2 ). We now develop X 2 ,

X 2 = (1 + δ 1 ) -1 n ℓ=1 E[z 1 t1,ℓℓ a * k T 1 a ℓ a * ℓ T 1,ℓ a ℓ a * ℓ (E j Q 1,ℓ )MQ 2 a k ] + (1 + δ 1 ) -1 j ℓ=1 E[z 1 t1,ℓℓ a * k T 1 a ℓ a * ℓ T 1,ℓ a ℓ r * ℓ (E j Q 1,ℓ )MQ 2 a k ] △ = U 1 + U 2 .
(2.4.38)

By formula (2.4.6), the term U 2 can be expressed as

U 2 = (1 + δ 1 ) -1 j ℓ=1 E[z 1 z 2 t1,ℓℓ q2,ℓℓ a * k T 1 a ℓ a * ℓ T 1,ℓ a ℓ r * ℓ (E j Q 1,ℓ )MQ 2,ℓ y ℓ y * ℓ Q 2,ℓ a k ] = j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ a * k T 1 a ℓ a * ℓ T 1,ℓ a ℓ E[r * ℓ (E j Q 1,ℓ )MQ 2,ℓ y ℓ y * ℓ Q 2,ℓ a k ] (1 + δ 1 ) + O(n -1/2 ).
We are in the framework of Remark 2.4.5. Write y ℓ y * ℓ = a ℓ a * ℓ + a ℓ r * ℓ + r ℓ r * ℓ + r ℓ a * ℓ , we will show that the only term which is not negligible is r ℓ a * ℓ . The term in a ℓ a * ℓ is :

E[r * ℓ (E j Q 1,ℓ )MQ 2,ℓ a ℓ a * ℓ Q 2,ℓ a k ] = 0.
Turning to the term a ℓ r * ℓ , we have

E|r * ℓ (E j Q 1,ℓ )MQ 2,ℓ a ℓ r * ℓ Q 2,ℓ a k | = O(n -1 ), hence j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ a * k T 1 a ℓ a * ℓ T 1,ℓ a ℓ E[r * ℓ (E j Q 1,ℓ )MQ 2,ℓ a ℓ r * ℓ Q 2,ℓ a k ] (1 + δ 1 ) ≤ K n j ℓ=1 |a * k T 1 a ℓ | ≤ K √ n j ℓ=1 |a * k T 1 a ℓ | 2 1/2 ≤ K √ n .
Moreover, for the term in r j r * j , we have

j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ a * k T 1 a ℓ a * ℓ T 1,ℓ a ℓ E[r * ℓ (E j Q 1,ℓ )MQ 2,l r ℓ r * ℓ Q 2,ℓ a k ] (1 + δ 1 ) = j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ a * k T 1 a ℓ a * ℓ T 1,ℓ a ℓ E{[r * ℓ (E j Q 1,ℓ )MQ 2,ℓ r ℓ -n -1 Tr(E j Q 1,ℓ )MQ 2,ℓ ]r * ℓ Q 2,ℓ a k } (1 + δ 1 ) ≤ K n j ℓ=1 |a * k T 1,ℓ a ℓ | = O(n -1/2 ).
The term in r ℓ a * ℓ is written as

j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ a * k T 1 a ℓ a * ℓ T 1,ℓ a ℓ E[r * ℓ (E j Q 1,ℓ )MQ 2,ℓ r ℓ a * ℓ Q 2,ℓ a k ] (1 + δ 1 ) (a) = j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ a * k T 1 a ℓ a * ℓ T 1,ℓ a ℓ E[r * ℓ (E j Q 1,ℓ )MQ 2,ℓ r ℓ ]E[a * ℓ Q 2,ℓ a k ] (1 + δ 1 ) + O(n -1/2 ) (b) =ψ M j j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ a * k T 1 a ℓ a * ℓ T 1,ℓ a ℓ E[a * ℓ Q 2,ℓ a k ] (1 + δ 1 ) + O(n -1/2 ),
where (a) is deduced from the fact that r * ℓ (E j Q 1,ℓ )MQ 2,ℓ r ℓ and a * ℓ Q 2,ℓ a k are uncorrelated (cf. Lemma 2.4.6) and (b) is from Lemma 2.4.3. The remaining term in the right hand side can be handled by the following lemma.

Lemma 2.4.7 Let (u) = (u n ) n∈N be a sequence of vectors with bounded Euclidean norms. Let (α ℓ ) 1≤ℓ≤n = (α ℓ,n ) 1≤ℓ≤n be an array of bounded real numbers. Then

j ℓ=1 α ℓ u * T 1 a ℓ E[a * ℓ Q 2,ℓ u] = - j ℓ=1 α ℓ u * T 1 a ℓ a * ℓ T 2 u z 2 t2,ℓℓ (1 + δ 2 ) + O(n -1/2 ).
The proof is postponed in Appendix 2.6.7.

Applying the precedent lemma with u = a k and α ℓ = z 1 z 2 t1,ℓℓ t2,ℓℓ a * ℓ T 1,ℓ a ℓ , we obtain

U 2 = -ψ M j j ℓ=1 z 1 t1,ℓℓ a * k T 1 a ℓ a * ℓ T 1,ℓ a ℓ a * ℓ T 2 a k (1 + δ 1 )(1 + δ 2 ) + O(n -1/2 ).
Gathering (2.4.36) and (2.4.38), and using the identity (2.4.10), we obtain

X = - n ℓ=1 z 1 t1,ℓℓ a * k T 1 a ℓ E[a * ℓ (E j Q 1,ℓ )MQ 2 a k ] -ψ M j j ℓ=1 z 1 t1,ℓℓ a * k T 1 a ℓ a * ℓ T 1,ℓ a ℓ a * ℓ T 2 a k (1 + δ 1 )(1 + δ 2 ) + O(n -1/2 ).
(2.4.39)

We now turn to the term Z in (2.4.35). By (2.4.9),

Z = -E[a * k T 1 (E j YY * Q 1 )MQ 2 a k ] = n ℓ=1 E[a * k T 1 (E j z 1 q1,ℓℓ y ℓ y * ℓ Q 1,ℓ )MQ 2 a k ] = z 1 n ℓ=1 t1,ℓℓ E[a * k T 1 (E j y ℓ y * ℓ Q 1,ℓ )MQ 2 a k ] + ǫ, where ǫ = - n ℓ=1 E[a * k T 1 (E j z(q 1,ℓℓ -t1,ℓℓ )y ℓ y * ℓ Q 1,ℓ )MQ 2 a k ] satisfies ǫ = O(n -1/2 ) (cf. the argument for ǫ 1 in (2.4.37)). Writing y ℓ y * ℓ = a ℓ a * ℓ + r ℓ r * ℓ + a ℓ r * ℓ + r ℓ a *
ℓ , we obtain :

Z = z 1 n ℓ=1 t1,ℓℓ a * k T 1 a ℓ E[a * ℓ (E j Q 1,ℓ )MQ 2 a k ] + z 1   j ℓ=1 t1,ℓℓ E[a * k T 1 r ℓ r * ℓ (E j Q 1,ℓ )MQ 2 a k ] + 1 n n ℓ=j+1 t1,ℓℓ E[a * k T 1 (E j Q 1,ℓ )MQ 2 a k ]   + z 1 j ℓ=1 t1,ℓℓ a * k T 1 a ℓ E[r * ℓ (E j Q 1,ℓ )MQ 2 a k ] + z 1 n ℓ=1 t1,ℓℓ E[a * k T 1 r ℓ a * ℓ (E j Q 1,ℓ )MQ 2 a k ] + O(n -1/2 ) △ = Z 1 + Z 2 + Z 3 + Z 4 + O(n -1/2 ).
(2.4.40) The term Z 1 cancels with the first term in the decomposition of X (cf. (2.4.39)). By developing Q 2 with Formula (2.4.6), the term Z 2 can be written as :

Z 2 =   z 1 j ℓ=1 t1,ℓℓ E[a * k T 1 r ℓ r * ℓ (E j Q 1,ℓ )MQ 2,ℓ a k ] + 1 n z 1 n ℓ=j+1 t1,ℓℓ E[a * k T 1 (E j Q 1,ℓ )MQ 2 a k ]   + j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ E[a * k T 1 r ℓ r * ℓ (E j Q 1,ℓ )MQ 2,ℓ y ℓ y * ℓ Q 2,ℓ a k ] + ǫ △ = W 1 + W 2 + ǫ, (2.4.41) where ǫ = O(n -1/2
) can be shown as (2.4.37). Consider first W 1 :

W 1 =   z 1 n j ℓ=1 t1,ℓℓ E[a * k T 1 (E j Q 1,ℓ )MQ 2,ℓ a k ] + z 1 n n ℓ=j+1 t1,ℓℓ E[a * k T 1 (E j Q 1,ℓ )MQ 2 a k ]   .
Denote Y 1:j = (y 1 , • • • , y j ). Writing :

(E j Q 1,ℓ )MQ 2,ℓ -(E j Q 1 )MQ 2 = (E j Q 1,ℓ )M(Q 2,ℓ -Q 2 ) + (E j Q 1,ℓ -E j Q 1 )MQ 2
and using (2.4.7) and (2.4.8), we have

z 1 n j ℓ=1 t1,ℓℓ E[a * k T 1 (E j Q 1,ℓ )M(Q 2,ℓ -Q 2 )a k ] = z 1 n j ℓ=1 t1,ℓℓ E[a * k T 1 (E j Q 1,ℓ )M(1 + y * j Q 2,j y j )Q 2 y j y * j Q 2 a k ] ≤ K n j ℓ=1 (E|1 + y * ℓ Q 2 y ℓ | 2 ) 1/2 (E|y * ℓ Q 2 a k | 2 ) 1/2 ≤ K n j ℓ=1 E|1 + y * ℓ Q 2 y ℓ | 2 1/2   n j=1 E|y * ℓ Q 2 a k | 2   1/2 ≤ K √ n (Ea * k Q 2 Y 1:j Y * 1:j Q * 2 a k ) 1/2 = O(n -1/2 ),
and the same argument applies to the term

(E j Q 1,ℓ -E j Q 1 )MQ 2 . Hence, W 1 = z 1 δ1 E[a * k T 1 (E j Q 1 )MQ 2 a k ] + O(n -1/2 ). (2.4.42)
Turning to W 2 and developing

y ℓ y * ℓ = a ℓ a * ℓ + a ℓ r * ℓ + r ℓ a * ℓ + r ℓ r * ℓ , for the term r ℓ a * ℓ , we have j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ E[a * k T 1 r ℓ r * ℓ (E j Q 1,ℓ )MQ 2,ℓ r ℓ a * ℓ Q 2,ℓ a k ] = j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ E a * k T 1 r ℓ r * ℓ (E j Q 1,ℓ )MQ 2,ℓ r ℓ - 1 n Tr(E j Q 1,ℓ )MQ 2,ℓ a * ℓ Q 2,ℓ a k ,
whose modulus is of order O(n -1/2 ). The estimate

j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ E[a * k T 1 r ℓ r * ℓ (E j Q 1,ℓ )MQ 2,ℓ a ℓ r * ℓ Q 2,ℓ a k ] = O(n -1/2 )
can be handled similarly and the term

j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ E[a * k T 1 r ℓ r * ℓ (E j Q 1,ℓ )MQ 2,ℓ a ℓ a * ℓ Q 2,ℓ a k ] = 1 n j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ E[a * k T 1 (E j Q 1,ℓ )MQ 2,ℓ a ℓ a * ℓ Q 2,ℓ a k ]
is bounded by Kn -1/2 . Finally,

W 2 = j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ E[a * k T 1 r ℓ r * ℓ (E j Q 1,ℓ )MQ 2,ℓ r ℓ r * ℓ Q 2,ℓ a k ] + O(n -1/2 ) (a) = j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ E[r * ℓ (E j Q 1,ℓ )MQ 2,ℓ r ℓ ]E[a * k T 1 r ℓ r * ℓ Q 2,ℓ a k ] + O(n -1/2 ) (b) = ψ M j a * k T 1 T 2 a k 1 n j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ + O(n -1/2 ), (2.4.43)
where (a) follows from the fact that r The term Z 3 satisfies

* ℓ (E j Q 1,ℓ )MQ 2,ℓ r ℓ and a * k T 1 r ℓ r * ℓ Q 2,
Z 3 = z 1 z 2 j ℓ=1 t1,ℓℓ t2,ℓℓ a * k T 1 a ℓ E[r * ℓ (E j Q 1,ℓ )MQ 2,ℓ y ℓ y * ℓ Q 2,ℓ a k ].
We are again in the framework of Remark 2.4.5. By developing y ℓ y * ℓ and relying on arguments as those already used, only the term with r j a * j is not negligible and we have :

Z 3 = z 1 z 2 j ℓ=1 t1,ℓℓ t2,ℓℓ a * k T 1 a ℓ E[r * ℓ (E j Q 1,ℓ )MQ 2,ℓ r ℓ a * ℓ Q 2,ℓ a k ] (a) = z 1 z 2 j ℓ=1 t1,ℓℓ t2,ℓℓ a * k T 1 a ℓ E[r * ℓ (E j Q 1,ℓ )MQ 2,ℓ r ℓ ]E[a * ℓ Q 2,ℓ a k ] (b) = ψ M j j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ a * k T 1 a ℓ E[a * ℓ Q 2,ℓ a k ] + O(n -1/2 ) (c) = -ψ M j j ℓ=1 z 1 t1,ℓℓ a * k T 1 a ℓ a * ℓ T 2 a k (1 + δ 2 ) + O(n -1/2 ), (2.4.44)
where (a) follows from the fact that r

* ℓ (E j Q 1,ℓ )MQ 2,ℓ r ℓ and a * ℓ Q 2,ℓ a k are uncorrelated, (b) is a consequence of Lemma 2.4.3 and (c) is from Lemma 2.4.7. Similarly, Z 4 = z 1 z 2 j ℓ=1 t1,ℓℓ t2,ℓℓ E[a * k T 1 r ℓ a * ℓ (E j Q 1,ℓ )MQ 2,ℓ y ℓ y * ℓ Q 2,ℓ a k ] + O(n -1/2 ) = a * k T 1 T 2 a k 1 n j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ E[a * ℓ (E j Q 1,ℓ )MQ 2,ℓ a ℓ ] + O(n -1/2 ) = a * k T 1 T 2 a k φ M j + O(n -1/2
Z = n ℓ=1 z 1 t1,ℓℓ a * k T 1 a ℓ E[a * ℓ (E j Q 1 )MQ 2 a k ] + z 1 δ1 E[a * k T 1 (E j Q 1 )MQ 2 a k ] + ψ M j a * k T 1 T 2 a k 1 n j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ -ψ M j j ℓ=1 z 1 t1,ℓℓ a * k T 1 a ℓ a * ℓ T 2 a k (1 + δ 2 ) + a * k T 1 T 2 a k φ M j + O(n -1/2 ).
Plugging this and (2.4.39) into (2.4.33), and noticing that -z tℓℓ (a

* ℓ T ℓ a ℓ (1 + δ) -1 + 1) = (1 + δ) -1 (cf. (2.4.10)), we obtain (2.4.31).
Step 2 : Expression of ψ M j In this section, we will show the following equation satisfied by ψ M j and ζ M kj :

ψ M j = 1 n TrT 1 MT 2 + ψ M j 1 n j ℓ=1 a * ℓ T 2 T 1 a ℓ (1 + δ 1 )(1 + δ 2 ) + γ n j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ + γφ M j + O(n -1/2 ).
(2.4.46)

Using identity (2.4.32), we obtain :

ψ M j = 1 n TrE[(E j Q 1 )MQ 2 ] = 1 n TrE[T 1 MQ 2 ] - z 1 δ1 n E[T 1 (E j Q 1 )MQ 2 ] + 1 n(1 + δ 1 ) TrE[T 1 AA * (E j Q 1 )MQ 2 ] - 1 n TrE[T 1 (E j YY * Q 1 )MQ 2 ] △ = 1 n TrT 1 MT 2 - z 1 δ1 n E[T 1 (E j Q 1 )MQ 2 ] + X + Z + ǫ, (2.4.47)
where X and Z are the last two terms and ǫ = O(n -1/2 ) by Theorem 2.4.1. X and Z can be treated in a similar way in the precedent section. More detailed, X satisfies

X = 1 n(1 + δ 1 ) n ℓ=1 E[a * ℓ (E j Q 1 )MQ 2 T 1 a ℓ ] = 1 n(1 + δ 1 ) n ℓ=1 E[a * ℓ (E j Q 1,ℓ )MQ 2 T 1 a ℓ ] + 1 n(1 + δ 1 ) n ℓ=1 E[z 1 t1,ℓℓ (E j a * ℓ Q 1,ℓ y ℓ y * ℓ Q 1,ℓ )MQ 2 T 1 a ℓ ] + O(n -1/2 ) = 1 n(1 + δ 1 ) n ℓ=1 E[a * ℓ (E j Q 1,ℓ )MQ 2 T 1 a ℓ ] + 1 n(1 + δ 1 ) n ℓ=1 E[z 1 t1,ℓℓ a * ℓ T 1,ℓ a ℓ a * ℓ (E j Q 1,ℓ )MQ 2 T 1 a ℓ ] + 1 n(1 + δ 1 ) j ℓ=1 E[z 1 t1,ℓℓ a * ℓ T 1,ℓ a ℓ r * ℓ (E j Q 1,ℓ )MQ 2 T 1 a ℓ ] + O(n -1/2 ).
By (2.4.10), we have

X = - 1 n n ℓ=1 E[z 1 t1,ℓℓ a * ℓ (E j Q 1,ℓ )MQ 2 T 1 a ℓ ] + 1 n(1 + δ 1 ) j ℓ=1 E[z 1 t1,ℓℓ a * ℓ T 1,ℓ a ℓ r * ℓ (E j Q 1,ℓ )MQ 2 T 1 a ℓ ] + O(n -1/2 ) (2.4.48)
and the second term satisfies

1 n(1 + δ 1 ) j ℓ=1 z 1 t1,ℓℓ a * ℓ T 1,ℓ a ℓ E[r * ℓ (E j Q 1,ℓ )MQ 2 T 1 a ℓ ] = 1 n(1 + δ 1 ) j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ a * ℓ T 1,ℓ a ℓ E[r * ℓ (E j Q 1,ℓ )MQ 2,ℓ y ℓ y * ℓ Q 2,ℓ T 1 a ℓ ] + O(n -1/2 ) (a) = 1 n(1 + δ 1 ) j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ a * ℓ T 1,ℓ a ℓ E[r * ℓ (E j Q 1,ℓ )MQ 2,ℓ r ℓ a * ℓ Q 2,ℓ T 1 a ℓ ] + O(n -1/2 ) (b) = ψ M j n j ℓ=1 a * ℓ T 1,ℓ a ℓ a * ℓ T 2 T 1 a ℓ (1 + δ 1 ) 2 (1 + δ 2 ) + O(n -1/2 ),
where (a) follows from Remark 2.4.5 and (b) follows from (2.4.11). The term Z can be expressed as :

Z = 1 n n ℓ=1 TrE[z 1 t1,ℓℓ T 1 (E j y ℓ y * ℓ Q 1,ℓ )MQ 2 ] + O(n -1/2 ) = 1 n n ℓ=1 E[z 1 t1,ℓℓ a * ℓ (E j Q 1,ℓ )MQ 2 T 1 a ℓ ] +   1 n j ℓ=1 E[z 1 t1,ℓℓ r * ℓ (E j Q 1,ℓ )MQ 2 T 1 r ℓ ] + 1 n n ℓ=j+1 1 n TrE[z 1 t1,ℓℓ (E j Q 1,ℓ )MQ 2 T 1 ]   + 1 n j ℓ=1 E[z 1 t1,ℓℓ y * ℓ (E j Q 1,ℓ )MQ 2 T 1 a ℓ ] + 1 n j ℓ=1 E[z 1 t1,ℓℓ a * ℓ (E j Q 1,ℓ )MQ 2 T 1 r ℓ ] + O(n -1/2 ) △ = Z 1 + Z 2 + Z 3 + Z 4 + O(n -1/2 ).
The term Z 1 cancels with the first term of X's decomposition in (2.4.48). The term Z 2 , Z 3 and Z 4 satisfy :

Z 2 = 1 n j ℓ=1 E[z 1 z 2 t1,ℓℓ t2,ℓℓ y * ℓ (E j Q 1,ℓ )MQ 2,ℓ y ℓ y * ℓ Q 2,ℓ T 1 r ℓ ] + z 1 δ 1 n TrE[T 1 (E j Q 1 )MQ 2 ] + O(n -1/2 ) = 1 n ψ M j TrT 1 T 2 1 n j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ + z 1 δ 1 n TrE[T 1 (E j Q 1 )MQ 2 ] + O(n -1/2 ),
and by Remark 2.4.5,

Z 3 = 1 n j ℓ=1 E[z 1 z 2 t1,ℓℓ t2,ℓℓ r * ℓ (E j Q 1,ℓ )MQ 2,ℓ y ℓ y * ℓ Q 2,ℓ T 1 a ℓ ] + O(n -1/2 ) = 1 n j ℓ=1 E[z 1 z 2 t1,ℓℓ t2,ℓℓ r * ℓ (E j Q 1,ℓ )MQ 2,ℓ r ℓ a * ℓ Q 2,ℓ T 1 a ℓ ] + O(n -1/2 ) = -ψ M j 1 n j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ a * ℓ T 2,ℓ T 1 a ℓ + O(n -1/2 ) = -ψ M j 1 n j ℓ=1 z 1 t1,ℓℓ a * ℓ T 2 T 1 a ℓ 1 + δ 2 + O(n -1/2 ),
where the last equality is from (2.4.11). For Z 4 , recall the definition of γ in (2.2.1),

Z 4 = 1 n j ℓ=1 E[z 1 z 2 t1,ℓℓ t2,ℓℓ a * ℓ (E j Q 1,ℓ )MQ 2,ℓ y ℓ y * ℓ Q 2,ℓ T 1 r ℓ ] + O(n -1/2 ) = 1 n TrT 1 T 2 1 n j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ E[a * ℓ (E j Q 1,ℓ )MQ 2,ℓ a ℓ ] + O(n -1/2 ) = γφ M j + O(n -1/2 ).
Plugging these terms in (2.4.47), by using (2.4.11) and (2.4.5), we obtain :

ψ M j = 1 n TrT 1 MT 2 + ψ M j 1 n j ℓ=1 a * ℓ T 2 T 1 a ℓ a * ℓ T 1 a ℓ (1 + δ 1 ) 2 (1 + δ 2 ) - z 1 t1,ℓℓ 1 + δ 2 + γ n j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ + γφ M j + O(n -1/2 ) = 1 n TrT 1 MT 2 + ψ M j 1 n j ℓ=1 a * ℓ T 2 T 1 a ℓ (1 + δ 1 )(1 + δ 2 ) + γ n j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ + γφ M j + O(n -1/2 ).
Step 3 : Relation between ζ M kj and θ M kj With the similar computations, in this section, we will show that :

ζ M kj = z 1 z 2 t1,kk t2,kk (1 + δ 1 )(1 + δ 2 )θ M kj + a * k T 1 a k a * k T 2 a k (1 + δ 1 )(1 + δ 2 ) ψ M j + O(n -1/2
ζ M kj = E[a * k (E j Q 1 )MQ 2 a k ] = E[a * k E j (Q 1,k + z 1 q1,kk Q 1,k y k y * k Q 1,k )M(Q 2,k + z 2 q2,kk Q 2,k y k y * k Q 2,k )a k ] = θ M kj + z 1 t1,kk E[a * k E j [Q 1,k y k y * k Q 1,k ]MQ 2,k a k ] + z 2 t2,kk E[a * k (E j Q 1,k )MQ 2,k y k y * k Q 2,k a k ] + z 1 z 2 t1,kk t2,kk E[a * k E j [Q 1,k y k y * k Q 1,k ]MQ 2,k y k y * k Q 2,k a k ] + O(n -1/2 ) △ = θ M kj + X 1 + X 2 + X 3 + O(n -1/2 ).
Using similar arguments as before, we get

X 1 = z 1 t1,kk a * k T 1,k a k E[a * k E j [Q 1,k ]MQ 2,k a k ] + O(n -1/2 ) = z 1 t1,kk a * k T 1,k a k θ M kj + O(n -1/2 ), X 2 = z 2 t2,kk a * k T 2,k a k θ M kj + O(n -1/2 ).
For k ≤ j,

X 3 = z 1 z 2 t1,kk t2,kk (a * k T 1,k a k )(a * k T 2,k a k )E[y * k (E j Q 1,k )MQ 2,k y k ] + O(n -1/2 ) = z 1 z 2 t1,kk t2,kk (a * k T 1,k a k )(a * k T 2,k a k )(θ M kj + ψ M j ) + O(n -1/2 ).
Using the identity (2.4.10), we finally obtain (2.4.49).

Step 4 : A system of perturbed linear equations.

Combining

z 1 z 2 t1,kk t2,kk θ M kj = a * k T 1 MT 2 a k (1 + δ 1 )(1 + δ 2 ) + ψ M j j ℓ=1 a * k T 1 a ℓ a * ℓ T 2 a k (1 + δ 1 ) 2 (1 + δ 2 ) 2 + a * k T 1 T 2 a k (1 + δ 1 )(1 + δ 2 ) 1 n j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ - a * k T 1 a k a * k T 2 a k (1 + δ 1 ) 2 (1 + δ 2 ) 2 + a * k T 1 T 2 a k (1 + δ 1 )(1 + δ 2 ) φ M j + O(n -1/2 ) (2.4.50) which implies that φ M j satisfies (1 -ν j )φ M j -(ω j + ν j η j )ψ M j = ν M j + O(n -1/2 )
where

ν j = 1 n j k=1 a * k T 1 T 2 a k (1 + δ 1 )(1 + δ 2 )
(2.4.51)

ν M j = 1 n j k=1 a * k T 1 MT 2 a k (1 + δ 1 )(1 + δ 2 )
(2.4.52)

η j = 1 n j ℓ=1 z 1 z 2 t1,ℓℓ t2,ℓℓ (2.4.53
)

ω j = 1 n j k=1 j ℓ=1,ℓ =k a * k T 1 a ℓ a * ℓ T 2 a k (1 + δ 1 ) 2 (1 + δ 2 ) 2 .
(2.4.54)

With these notations, Equation (2.4.46) is rewritten as

-γφ M j + (1 -ν j -γη j )ψ M j = γ M + O(n -1/2 )
,

where γ M = 1 n TrT 1 MT 2 .
We end up with a system of two perturbed linear equations in φ M j and ψ M j :

   (1 -ν j )φ M j -(ω j + ν j η j )ψ M j = ν M j + O(n -1/2 ) -γφ M j + (1 -ν j -γη j )ψ M j = γ M + O(n -1/2 ).
(2.4.55)

The determinant of the system is

∆ j = (1 -ν j ) 2 -γη j -γω j ,
which has several interesting characters.

Lemma 2.4.8 We have :

1. The determinant of the system ∆ j coincides with ∆ n in (2.2.7) when j = n.

The sequence

(∆ n ) satisfies for all z 1 , z 2 ∈ C + and z 1 = z 2 , 1 ∆ n = s n (z 1 ) -s n (z 2 ) z 1 -z 2 ,
where

s n (z) = z(1 + δ n )(1 + δn ). 3. For all z 1 , z 2 ∈ C + , lim inf n inf j∈{1,••• ,n} |∆ j (z 1 , z 2 )| > 0.
The proof is in Appendix 2.6.8. With the precedent lemma, the system has the solution

φ M j ψ M j = 1 ∆ j ν M j (1 -ν j -γη j ) + γ M (ω j + ν j η j ) γν M j + γ M (1 -ν j ) + O(n -1/2 ).
(2.4.56)

Now we take M = I N and the convention that ν 0 = ω 0 = η 0 = 0. Noticing that ν I N j = ν j and γ I N = γ, and replacing the solution into (2.4.50), we obtain :

z 1 z 2 t1,jj t2,jj θ I N jj (z 1 , z 2 ) n =(ν j -ν j-1 ) + 1 2 (ω j -ω j-1 ) + η j (ν j -ν j-1 ) ψ I N j (z 1 , z 2 ) + (ν j -ν j-1 )φ I N j (z 1 , z 2 ) + O(n -3/2 ) =(ν j -ν j-1 ) + 1 2 γ(ω j -ω j-1 ) + γη j (ν j -ν j-1 ) ∆ j + (ν j -ν j-1 )(ν j (1 -ν j ) + γν j ) ∆ j + O(n -3/2 ).
(2.4.57)

Since z 1 , z 2 are symmetric, we can achieve the same equation by interchanging z 1 and z 2 . Moreover, by diagonalization,

T 1 T 2 = T 2 T 1 , which implies ν j (z 1 , z 2 ) = ν j (z 2 , z 1 ). Thus, we get z 1 z 2 t1,jj t2,jj θ I N jj (z 2 , z 1 ) n = z 1 z 2 t1,jj t2,jj θ I N jj (z 1 , z 2 ) n + O(n -3/2 )
which leads to

1 n z 1 z 2 t1,jj t2,jj (ψ I N j (z 1 , z 2 ) + θ I N jj (z 1 , z 2 ) + θ I N jj (z 2 , z 1 )) = 1 n γ(η j -η j-1 ) ∆ j + 2 n z 1 z 2 t1,jj t2,jj θ I N jj + O(n -3/2 ) = 2(ν j -ν j-1 )(1 -ν j ) + γ(η j -η j-1 ) + γ(ω j -ω j-1 ) ∆ j + O(n -3/2 ).
On the other hand, taking ∆ 0 = 1 and noticing that

∆ j-1 -∆ j = 2(ν j -ν j-1 )(1 -ν j ) + γ(η j -η j-1 ) + γ(ω j -ω j-1 ) + O(n -2
), then we have

1 n z 1 z 2 t1,jj t2,jj (ψ I N j (z 1 , z 2 ) + θ I N jj (z 1 , z 2 ) + θ I N jj (z 2 , z 1 )) = ∆ j-1 -∆ j ∆ j + O(n -3/2 ). (2.4.58)
Since for all j ∈ {1, • • • , n},

1 n z 1 z 2 t1,jj t2,jj (ψ I N j (z 1 , z 2 ) + θ I N jj (z 1 , z 2 ) + θ I N jj (z 2 , z 1 )) ≤ K n , then ∆ j-1 -∆ j ∆ j ≤ K n .
(2.4.59)

Now we introduce the logarithm function for complex number. For z ∈ C * , denote

log(z) = log |z| + i arg(z), arg(z) ∈ [-π, π).
We can verify easily that log(1 + z) = z + o(|z|) when |z| → 0 and d dz log(1 + z) = 1 1+z . Combining (2.4.58) and (2.4.59), we have

1 n n j=1 z 1 z 2 t1,jj t2,jj (ψ I N j (z 1 , z 2 ) + θ I N jj (z 1 , z 2 ) + θ I N jj (z 2 , z 1 )) = n j=1 ∆ j-1 -∆ j ∆ j + O(n -1/2 ) = n j=1 log 1 + ∆ j-1 -∆ j ∆ j + O(n -1/2 ) = n j=1 log ∆ j-1 ∆ j + O(n -1/2
ξ 3j - n j=1 log ∆ j-1 ∆ j ---→ n→∞ 0, max    n j=1 ξ 3j , n j=1 log ∆ j-1 ∆ j    ≤ K,
then by Lemma 2.4.1, with the same argument as Proposition 2.4.1,we have

∂ ∂z 1 n j=1 ξ 3j - ∂ ∂z 1 n j=1 log ∆ j-1 ∆ j ---→ n→∞ 0,
and

∂ ∂z 1 n j=1 ξ 3j - ∂ ∂z 1 n j=1 log ∆ j-1 ∆ j ≤ K.
By another application of Lemma 2.4.1, we have

∂ 2 ∂z 1 ∂z 2 n j=1 ξ 3j - ∂ 2 ∂z 1 ∂z 2 n j=1 log ∆ j-1 ∆ j ---→ n→∞ 0. (2.4.61)
By a simple calculation,

∂ ∂z 1 n j=1 log ∆ j-1 ∆ j = n j=1 1 ∆ j-1 ∂∆ j-1 ∂z 1 - 1 ∆ j ∂∆ j ∂z 1 = - 1 ∆ n ∂∆ n ∂z 1 .
This along with (1) and (2) in Lemma 2.4.8 conclude the proof of Lemma 2.4.5.

A similar result holds true for n j=1 ξ 4j . Lemma 2.4.9 Recall the expression of n j=1 ξ 4j in (2.4.29). We have

∂ 2 ∂z 1 ∂z 2 n j=1 ξ 4j -Θ 1,n P ---→ n→∞ 0 where Θ 1,n is defined in (2.2.

13).

The same method as the proof of Lemma 2.4.5 applies for this lemma. We just recall some elements of the proof. We first calculate a * j (E j Q j (z 1 ))(E j Q T j (z 2 ))ā j and denote :

ψ j (z 1 , z 2 ) = 1 n TrE{[E j Q(z 1 )]E j Q T (z 2 )} = 1 n TrE{[E j Q(z 1 )]Q T (z 2 )}, ζ kj (z 1 , z 2 ) = E{a * k [E j Q(z 1 )]E j Q T (z 2 )ā k } = E{a * k [E j Q(z 1 )]Q T (z 2 )ā k }, θ kj (z 1 , z 2 ) = E{a * k [E j Q k (z 1 )]E j Q T k (z 2 )ā k } = E{a * k [E j Q k (z 1 )]Q T k (z 2 )ā k }, φ j = 1 n j k=1 z 1 z 2 tkk (z 1 ) tkk (z 2 )θ kj .
Similar derivations as the proof of Lemma 2.4.5 yield the perturbed system :

   (1 -ϑν † j )φ j -( θω † j + |ϑ| 2 ν † j η j )ψ j = ν † j + O(n -1/2 ) -ϑγ † φ j + (1 -θν † j -|ϑ| 2 γ † η j )ψ j = γ † + O(n -1/2 ).
(2.4.62)

where

ν † j = 1 n j k=1 a * k T 1 T T 2 āk (1 + δ 1 )(1 + δ 2 ) ν † j = 1 n j k=1 a T k T T 1 T 2 a k (1 + δ 1 )(1 + δ 2 ) γ † = 1 n TrT 1 T T 2 ω † j = 1 n j k=1 j ℓ=1,ℓ =k a * k T 1 a ℓ a T ℓ T T 2 āk (1 + δ 1 ) 2 (1 + δ 2 ) 2 .
We have similar proprieties for the determinant of the system

∆ ϑ j (z 1 , z 2 ) = (1 -ϑν † j )(1 -θν † j ) -|ϑ| 2 γ † (η j + ω † j ).
Lemma 2.4.10 For all z 1 , z 2 ∈ C + , We have 1. The determinant of the system ∆ ϑ j coincides with ∆ ϑ n in (2.2.8) when j = n.

For all

z 1 , z 2 ∈ C + , lim inf n inf j∈{1,••• ,n} |∆ j (z 1 , z 2 )| > 0.
The proof is in Appendix 2.6.9.

Then the resolution of the system yields that

φ j ψ j = 1 ∆ ϑ j ν † j (1 -θν † j ) + θγ † ω † j γ † + O(n -1/2 ), (2.4.63)
The other term in

n j=1 ξ 4j is a T j (E j Q T j (z 1 ))(E j Q j (z 2 ))a j . With Remark 2.2.3, we also consi- der ψj (z 1 , z 2 ) = 1 n TrE{[E j Q T (z 1 )]E j Q(z 2 )} = 1 n TrE{[E j Q T (z 1 )]Q(z 2 )}, ζkj (z 1 , z 2 ) = E{a T k [E j Q T (z 1 )]E j Q(z 2 )a k } = E{a T k [E j Q T (z 1 )]Q(z 2 )a k }, θkj (z 1 , z 2 ) = E{a T k [E j Q T k (z 1 )]E j Q k (z 2 )a k } = E{a * k [E j Q T k (z 1 )]Q k (z 2 )ā k }, φj = 1 n j k=1 z 1 z 2 tkk (z 1 ) tkk (z 2 ) θkj .
Then we have the perturbed system

   (1 -θν † j ) φj -(ϑω † j + |ϑ| 2 ν † j η j ) ψj = ν † j + O(n -1/2 ) -θγ † φj + (1 -ϑν † j -|ϑ| 2 γ † η j ) ψj = γ † + O(n -1/2 ).
(2.4.64)

The determinant of this system is again ∆ ϑ j and the solution of the system is

φj ψj = 1 ∆ ϑ j ν † j (1 -ϑν † j ) + ϑγ † ω † j γ † + O(n -1/2 ), (2.4.65)
Remark 2.4.6 Notice that in [START_REF] Hachem | A CLT for informationtheoretic statistics of non-centered Gram random matrices[END_REF], when

z 1 , z 2 ∈ R, ν † j = ν † j
which is coherent with their result.

Then it suffices to show that

∂ 2 ∂z 1 ∂z 2 1 n n j=1 z 1 z 2 tjj (z 1 ) tjj (z 2 )(|ϑ| 2 ψ j + ϑθ jj + θθ jj ) -Θ 1,n ---→ n→∞ 0. Since 1 n n j=1 (z 1 z 2 tjj (z 1 ) tjj (z 2 )|ϑ| 2 ψ j + z 1 z 2 tjj (z 1 ) tjj (z 2 )(ϑθ jj + θθ jj ) = n j=1 ∆ ϑ j-1 -∆ ϑ j ∆ ϑ j + O(n -1/2 ) = n j=1 log ∆ ϑ j-1 ∆ ϑ j + O(n -1/2 ),
with a similar application of Lemma 2.4.1, Lemma 2.4.9 is proved. Gathering Proposition 2.4.1 , Lemma 2.4.2, 2.4.4, 2.4.5 and 2.4.9, we achieve the expression (2.2.11) of the covariance between M n (z 1 ) and M n (z 2 ). This ends the computation of the covariance.

Proof of Theorem 2.2.2 : Computation of the bias

In this section, we will calculate the bias

L 2 n (f ) = E( N i=1 f (λ i )) -N R f (x)δ n (x)dx.
This issue has been studied in [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF][START_REF] Hachem | A CLT for informationtheoretic statistics of non-centered Gram random matrices[END_REF][START_REF] Vallet | Improved subspace estimation for multivariate observations of high dimension : The deterministic signals case[END_REF][START_REF] Haagerup | A new application of random matrices : Ext(C * red (F 2 )) is not a group[END_REF].

Theorem 2.5.1 Under Assumptions 2.2.1 and 2.2.2, we have

L 2 n (f ) -B n (f ) ---→ n→∞ 0,
where B n (f ) is defined in (2.2.16).

Proof : We will show that

1. For all z ∈ C + , |M 2 n (z) -B(z)| -----→ N,n→∞ 0 2. M 2 n (z) -B(z)
forms an equicontinuous family for z ∈ C n . From (3.1.2) and the relation that δ n -δn = 1-cn z ,

δ n = 1 n Tr -z(1 + δ n - 1 -c n z )I N + AA * 1 + δ n -1
.

(2.5.1)

Then by replacing δ n by 1 n TrEQ, we consider

A n = 1 n TrEQ - 1 n Tr -z 1 + 1 n TrEQ - 1 -c n z I N + AA * 1 + 1 n TrEQ -1 . (2.5.2)
Subtracting (2.5.1) by (2.5.2), we have

TrEQ -nδ n = nA n 1 - z n TrT 2 - 1 n Tr TAA * T (1 + δ n ) 2 -1
+ o(1).

(2.5.3) Now we will calculate nA n . To simplify the notations, we denote

S = -z(1 + 1 n TrEQ - 1 -c n z )I N + AA * 1 + 1 n TrEQ . (2.5.4) By A -1 -B -1 = -B -1 (A -B)A -1 , we have nA n = -ETrS -1 [ n j=1 y j y * j -zI N -S](YY * -zI N ) -1 .
(2.5.5)

From YY * -zI N + zI N = n j=1 y j y * j , taking the inverse of YY * -zI N from both sides along with (2.4.9), we have

I N + z(YY * -zI N ) -1 = n j=1 β j y j y * j Q j .
Taking trace on both sides and dividing by n, we have

c n + z n Tr(YY * -zI N ) -1 = 1 n n j=1 β j y * j Q j y j = 1 - 1 n n j=1 β j . Thus 1 n Tr(Y * Y -zI n ) -1 = - 1 nz n j=1 β j .
Plugging the precedent equality and (2.4.9) into (2.5.5), we have

nA n = -E n j=1 {β j y * j Q j S -1 y j - 1 n Eβ j ETrS -1 Q} + n j=1 1 1 + 1 n TrEQ Ea * j QS -1 a j . (2.5.6)
Then by (2.4.6), we have

ETrS -1 Q -ETrS -1 Q j = -ETrβ j S -1 Q j y j y * j Q j = -b j E(1 -β j γ j )y * j Q j S -1 Q j y j . (2.5.7)
By Proposition 2.3.2,

|Eβ j γ j y * j Q j S -1 Q j y j | ≤ K √ n .
Then

ETrS -1 Q -ETrS -1 Q j = -b j Ey * j Q j S -1 Q j y j + o(1). Since β j = b j -b 2 j γ j + β j b 2 j γ 2 j , we have n j=1 {Eβ j y * j Q j S -1 y j - 1 n Eβ j ETrS -1 Q j -Eβ j a * j Q j S -1 a j } = n j=1 -b 2 j Eγ j y * j Q j S -1 y j + b 2 j Eγ j a * j Q j S -1 a j + n j=1 b 2 j [Eβ j γ 2 j y * j Q j S -1 y j -Eβ j γ 2 j a * j Q j S -1 a j - 1 n Eβ j γ 2 j ETrS -1 Q j ] = n j=1 -b 2 j Eγ j y * j Q j S -1 y j + b 2 j Eγ j a * j Q j S -1 a j + n j=1 b 2 j [Eβ j γ 2 j y * j Q j S -1 y j -Eβ j γ 2 j a * j Q j S -1 a j - 1 n Eβ j γ 2 j TrS -1 Q j ] + n j=1
Cov(β j γ 2 j ,

1 n TrS -1 Q j ).
By Lemma 2.3.3, we have

n j=1 b 2 j [Eβ j γ 2 j y * j Q j S -1 y j -Eβ j γ 2 j a * j Q j S -1 a j - 1 n Eβ j γ 2 j TrS -1 Q j ] ≤ Kη 2 n and n j=1
Cov(β j γ 2 j ,

1 n TrS -1 Q j ) ≤ Kη 2 n .
Then

n j=1 Eγ j y * j Q j S -1 y j -Eγ j a * j Q j S -1 a j = n j=1 E y * j Q j y j -n -1 TrQ j -a * j Q j a j y * j Q j S -1 y j - 1 n TrQ j S -1 -a * j Q j S -1 a j + n j=1 Cov 1 n TrQ j + a j Q j a j , 1 n TrQ j S -1 .
(2.5.8)

From Lemma 2.3.2, the second term is of the order 1 N . Now we will show that

n j=1 1 1 + 1 n TrEQ Ea * j QS -1 a j - n j=1 Eβ j a * j Q j S -1 a j = - n j=1 b 2 j n Ea * j Q j Q j S -1 a j + o(1). (2.5.9)
From the relation that

Q = Q j -β j Q j y j y * j Q j , we have n j=1 Ea * j QS -1 a j = n j=1 Ea * j Q j S -1 a j - n j=1 Eβ j a * j Q j y j y * j Q j S -1 a j (a) = n j=1 Ea * j Q j S -1 a j - n j=1 b j n Ea * j Q j Q j S -1 a j - n j=1 b j Ea * j Q j a j a * j Q j S -1 a j + o(1) (b) = n j=1 Ea * j Q j S -1 a j - n j=1 b j n Ea * j Q j Q j S -1 a j - n j=1 b j Ea * j Q j a j Ea * j Q j S -1 a j + o(1)
where (a) follows from Eβ j = b j + o(1) and (b) is using the fact that a * j Q j a j and a * j Q j S -1 a j are uncorrelated. From the definition of b j , (cf. (2.3.9)),

1 1 + 1 n TrEQ -b j = b j Ea * j Q j a j 1 + 1 n TrEQ + o(1),
Then we have

n j=1 1 1 + 1 n TrEQ j Ea * j Q j S -1 a j - n j=1 Eβ j a * j QS -1 a j - n j=1 b j 1 + 1 n TrEQ j Ea * j Q j a j Ea * j Q j S -1 a j = n j=1 1 1 + 1 n TrEQ j -b j Ea * j Q j S -1 a j - n j=1 b j 1 + 1 n TrEQ j Ea * j Q j a j Ea * j Q j S -1 a j + o(1)
= o(1).

(2.5.10)

Moreover n j=1 b j n(1 + 1 n TrEQ j ) Ea * j Q j Q j S -1 a j - n j=1 b 2 j n Ea * j Q j Q j S -1 a j = n j=1 b j n(1 + 1 n TrEQ j ) Ea * j Q j Q j S -1 a j - n j=1 b 2 j n Ea * j Q j Q j S -1 a j = n j=1 b 2 j Ea * j Q j a j n(1 + 1 n TrEQ j ) Ea * j Q j Q j S -1 a j + o(1) = n j=1 b 2 j n(1 + 1 n TrEQ j ) Ea * j Q j a j a * j Q j Q j S -1 a j - n j=1 b 2 j n(1 + 1 n TrEQ j ) Cov a * j Q j a j , a * j Q j Q j S -1 a j + o(1).
(2.5.11)

From Lemma 2.3.1, the second term is of the order O 1 n and the first term is also of the order

O 1 n as EQ j a j a * j Q j Q j S -1 = O(n -1 ).
Thus, (2.5.10) and (2.5.11) imply (2.5.9). Since for all matrices M 1 , M 2 with bounded spectral measures,

1 n TrM 1 S -1 M 2 - 1 n TrM 1 TM 2 -----→ N,n→∞ 0,
gathering (2.5.6), (2.5.7), (2.5.8) and (2.5.9), we have

nA n = - n j=1 b 2 j n 2 ETrQ j TQ j - n j=1 b 2 j n Ea * j Q j TQ j a j - n j=1 b 2 j n Ea * j Q j Q j Ta j + n j=1 b 2 j E y * j Q j y j -n -1 TrQ j -a * j Q j a j × y * j Q j Ty j - 1 n TrQ j T -a * j Q j Ta j + o(1
nA n = n j=1 |ϑ| 2 b 2 j n 2 ETrQ j TQ T j + n j=1 ϑb 2 j n Ea * j Q j TQ T j āj + n j=1 θb 2 j n Ea T j Q T j Q j Ta j + κ n j=1 b 2 j n 2 E N i=1 [Q j ] ii [Q j T ] ii + o(1)
.

By Lemma 2.4.1, 1 n E N i=1 [Q j ] ii [Q j T ] ii - 1 n N i=1 t ii [T 2 ] ii ---→ n→∞ 0.
Hence, as b j = -z tjj + o(1), the term on κ is

κ n j=1 b 2 j n 2 E N i=1 [Q j ] ii [Q j T ] ii = κz 2 n 2 n j=1 t2 jj N i=1 t ii [T 2 ] ii + o(1)
.

Now we will calculate 1 n n j=1
Eb 2 j TrQ j TQ T j .

From the argument in the proof of Lemma 2.4.4 and 2.4.9, consider

ψ T j (z 1 , z 2 ) = 1 n TrE{[E j Q(z 1 )]TE j Q T (z 2 )} = 1 n TrE{[E j Q(z 1 )]TQ T (z 2 )}, ζ T kj (z 1 , z 2 ) = E{a * k [E j Q(z 1 )]TE j Q T (z 2 )ā k } = E{a * k [E j Q(z 1 )]TQ T (z 2 )ā k }, θ T kj (z 1 , z 2 ) = E{a * k [E j Q k (z 1 )]TE j Q T k (z 2 )ā k } = E{a * k [E j Q k (z 1 )]TQ T k (z 2 )ā k }, φ T j = 1 n j k=1 z 1 z 2 tkk (z 1 ) tkk (z 2 )θ T kj .
The perturbed system is :

   (1 -ϑν † j )φ T j -( θω † j + |ϑ| 2 ν † j η j )ψ T j = ν † j (T) + O(n -1/2 ) -ϑγ † φ T j + (1 -θν † j -|ϑ| 2 γ † η j )ψ T j = γ † T + O(n -1/2 ).
(2.5.12)

where

ν † j (T) = 1 n j k=1 a * k T 1 TT T 2 āk (1 + δ 1 )(1 + δ 2 ) γ † T = 1 n TrT 1 TT T 2 .
The determinant of this system is ∆ ϑ j and the resolution of the system yields that

φ T j ψ T j = 1 ∆ ϑ j F T j (1 -θν † j -|ϑ| 2 γ † η j ) + γ T j ( θω † j + |ϑ| 2 ν † j η j ) ϑγ † ν † j (T) + γ † T (1 -ϑν † j ) + O(n -1/2 ).
(2.5.13)

Then we have

1 n 2 n j=1 Eb 2 j TrQ j TQ T j = 1 n 2 n j=1 Eb 2 j TrQTQ T + o(1) = z 2 n 2 ψ T n (z, z) n j=1 t2 jj + o(1)
.

Now we will calculate n j=1 θb 2 j n Ea T j Q T j Q j Ta j . We first show that n j=1 θb 2 j n Ea T j Q T j Q j Ta j -Ea T j Q T j TQ j a j ---→ n→∞ 0.
By Lemma 2.4.3, it suffices to prove that

n j=1 θb 2 j n Ea T j Q T QTa j -Ea T j Q T TQa j ---→ n→∞ 0.
(2.5.14)

We have

n j=1 θb 2 j n Ea T j Q T QTa j - n j=1 θb 2 j n Ea T j Q T TTa j ≤ K n n j=1 Ea T j Q T (QT -TT)a j (a) ≤ K n n j=1 E (QT -TT)a j (b) = o(1),
where (a) follows from the fact that Q ≤ (Imz) -1 and (b) is from (2.4.14) in Theorem 2.4.1.

Same result holds true for

θb 2 j n Ea T j Q T TQa j : n j=1 θb 2 j n Ea T j Q T TTa j - n j=1 θb 2 j n Ea T j Q T TQa j = o(1).
This proves (2.5.14).

Then it suffices to consider

ψT j (z 1 , z 2 ) = 1 n TrE[E j Q T (z 1 )TE j Q(z 2 )] = 1 n TrE[E j Q T (z 1 )TQ(z 2 )], ζT kj (z 1 , z 2 ) = E[a T k E j Q T (z 1 )TE j Q(z 2 )a k ] = E[a T k E j Q T (z 1 )TQ(z 2 )a k ], θT kj (z 1 , z 2 ) = E[a T k E j Q T k (z 1 )TE j Q k (z 2 )a k ] = E[a T k E j Q T k (z 1 )TQ k (z 2 )a k ], φT j = 1 n j k=1 z 1 z 2 tkk (z 1 ) tkk (z 2 ) θT kj .
We have the solution

  φT j ψT j   = 1 ∆ ϑ j ν † j (T)(1 -ϑν † j -|ϑ| 2 γ † η j ) + γT (ϑω † j + |ϑ| 2 ν † j η j ) θγ † ν † j (T) + γ † T (1 -θν † j ) + O(n -1/2 ), (2.5.15)
where

ν † j (T) = 1 n j k=1 a T k T T 1 TT 2 a k (1 + δ 1 )(1 + δ 2 ) γ † T = 1 n TrT T 1 TT 2 .
Then with (2.5.13) and (2.5.15), we have

|ϑ| 2 n n j=1 Eb 2 j TrQ j TQ T j + n j=1 ϑb 2 j n Ea * j Q j TQ T j āj + n j=1 θb 2 j n Ea T j Q T j Q j Ta j =|ϑ| 2 ω(z, z)ψ T n (z, z) + ϑφ T n (z, z) + θφ T n (z, z) = 1 ∆ ϑ n |ϑ| 2 η(ϑγ † ν † T + γ † T (1 -ϑν † )) + ϑν † T (1 -θν † -|ϑ| 2 γ † η) +ϑγ † T ( θω † + |ϑ| 2 ν † η) + θν † T (1 -ϑν † -|ϑ| 2 γ † η) + θγ T (ϑω † + |ϑ| 2 ν † η) + o(1),
where all terms are evaluated on (z 1 , z 2 ) = (z, z).

Combining this with (2.5.3) and recall the definition of Υ n in (2.2.9), and η n + ω † n = γ † , we achieve the final expression of the bias in (2.2.16).

The proof is completed by showing that E(TrQ)nδ n forms an equicontinuous family for z ∈ C n . With the precedent work, it suffices to show that L ′ n (z) is bounded for z ∈ C n where

L n (z) = n j=1 b 2 j Eγ j y * j Q j S -1 y j - 1 n TrQ j S -1 -a * j Q j S -1 a j .
The terms in L ′ n (z) are linear combinations of

L 1 n (z) = 2b j b ′ j Eγ j y * j Q j S -1 y j - 1 n TrQ j S -1 -a * j Q j S -1 a j L 2 n (z) = b 2 j Eγ ′ j y * j Q j S -1 y j - 1 n TrQ j S -1 -a * j Q j S -1 a j L 3 n (z) = b 2 j Eγ j y * j Q 2 j S -1 y j - 1 n TrQ 2 j S -1 -a * j Q 2 j S -1 a j L 4 n (z) = -b 2 j Eγ j y * j Q j S -2 S ′ y j - 1 n TrQ j S -2 S ′ -a * j Q j S -2 S ′ a j .
According to Lemma 2.3.5, all terms are indeed bounded. This concludes the computations of the bias.

Appendices

Proof of Proposition 2.3.1

We begin the proof with the replacement of the entries x ij satisfying that there exists a sequence

η n → 0 such that η n n 1/5 → ∞, |x ij | < η n √ n. By Assumption 2.
2.1, we may select

η n ↓ 0 such that : 1 η 4 n E|x 11 | 4 I(|x 11 | ≥ η n √ n) -----→ N,n→∞ 0. Let Y n = ( 1 √ n X n + A n ) with X n , an N × n matrix with the entries xij = x ij I |x ij |≤ηn √ n . We have then P(Y n Y * n = Y n Y * n ) ≤ ij P(|x ij | ≥ η n √ n) ≤ 1 η 4 E|x 11 | 4 I(|x 11 | ≥ η n √ n) = o(1).
(2.6.1)

Denote Y n = ( 1 √ n X n + A n ) with the entries xij = (x ij -Ex ij )/σ ij , where σ 2 ij = E|x ij -Ex ij | 2 .
We will show that

E(x 2 11 ) ---→ n→∞ ϑ, E|x 11 | 4 ---→ n→∞ E|x 11 | 4 , and
Trf ( Y n Y * n ) = Trf (Y n Y * n ) + o P (1). (2.6.2) Since E|x 11 | 2 < ∞, in particular, E|x 11 | < ∞. By dominated convergence theorem, |Ex 11 | ---→ n→∞ 0, |σ 11 -1| ---→ n→∞ 0, |Ex 2 11 -x 2 11 | ---→ n→∞ 0, (2.6.3)
which implies that

E(x 2 11 ) -ϑ = E x11 -Ex 11 σ 11 2 -Ex 2 11 ---→ n→∞ 0.
Same method yields that

E|x 11 | 4 ---→ n→∞ E|x 11 | 4 .
We shall use the following proposition (cf. [START_REF] Bai | Spectral analysis of large dimensional random matrices[END_REF]Corollary A.42]) to show (2.6.2).

Proposition 2.6.1 Let A and B be two N × n matrices and the empirical estimate distribution of S A = AA * and S B = BB * be denoted by F S A and F S B . Then,

d 4 LP (F S A , F S B ) ≤ 2 N 2 (Tr(AA * + BB * ))(Tr[A -B][A -B] * ). As 1 √ n X n + A n 1 √ n X n + A n * ≤ 2 1 n X n X * n + A n A * n , 2
denote λ M max , the largest eigenvalue of the matrix M, then we get

lim sup λ Yn Y * n max ≤ 2 lim sup λ 1 n Xn X * n max + 2λ AnA * n max .
From [5, Theorem 5.11], λ

G n (x) = N × (F Y Y * -F n ) and G n (x) = N (F Y Y * -F n ) and let λ A
i denote the i-th smallest eigenvalue of A. Using Proposition 2.6.1, for an analytic function f which satisfies Assumption 2.2.3, we have,

E f (x)d G n (x) -f (x)d G n (x) (2.6.4) ≤ E N j=1 f ( λj ) -f ( λj ) ≤ KE N j=1 λj -λj ≤ 2K n E 1/2 Tr( X n -X n )( X n -X n ) * E 1/2 Tr[ Y Y * + Y Y * ] .
where K is the bound of |f ′ (x)| in the compact which contains both supports of Y Y * and Y Y * .

From the definition of xij ,

|E|x 11 | 2 -1| 2 = (E|x 11 | 2 I |x 11 |≥ηn √ n ) 2 ≤ 1 η 4 n n 2 (E|x 11 | 4 I |x 11 |≥ηn √ n ) 2 = o(n -2 ).
Moreover,

|Ex 11 | 2 ≤ 1 n 3 η 6 n E|x 11 | 4 I(|x 11 | ≥ η n √ n) 2 = o(n -2 ).

These give us

ETr(

X n -X n )( X n -X n ) (2.6.5) ≤ 2 ij [(1 -σ -1 ij ) 2 E|x ij | 2 + σ -2 ij |Ex ij | 2 ] ≤ 2 ij [(1 -σ -1 ij ) 2 + |Ex ij | 2 ] = o(1).
Similarly,

1 n E Tr Y Y * + Y Y * (2.6.6) ≤ 1 n ij E 1 √ n xij + a ij 2 + 1 √ n xij + a ij 2 ≤ 2 n ij E 1 n |x ij | 2 + 1 n |x ij | 2 + 2 |a ij | 2 ≤ K.
Plugging the estimations (2.6.5) and (2.6.6) into (2.6.4), along with (2.6.1), we conclude that

Trf ( Y Y * ) = Trf (YY * ) + o P (1).

Proof of Proposition 2.3.2

It has been shown that the largest value of the limiting support S of the eigenvalues of [START_REF] Bai | No Eigenvalues Outside the Support of the Limiting Spectral Distribution of Large Dimensional Sample Covariance Matrices[END_REF][START_REF] Yin | On the limit of the largest eigenvalue of the large dimensional sample covariance matrix[END_REF]). Let the contour C n , ε n be defined in Section 2.3.2 and µ ℓ < 0 and µ r > ℓ + . The partition of C n is identical to that used in [11, Section 1]. With probability one (see [START_REF] Bai | No Eigenvalues Outside the Support of the Limiting Spectral Distribution of Large Dimensional Sample Covariance Matrices[END_REF] and [START_REF] Bai | Exact Separation of Eigenvalues of Large Dimensional Sample Covariance Matrices[END_REF]), for all ǫ > 0, lim sup

1 n X n X * n is (1 + √ c) 2 (cf.
λ∈eig(YnY * n ) d(λ, S n ) < ǫ where d(x, S) is the Euclidean distance of x to the set S, eig(Y n Y * n ) denotes the set of eigenvalues of Y n Y *
n and S n is the support associated with F n . Then with probability one, for all N large, as f is continuous over C,

f (z) M n (z) -Mn (z) dz ≤ 4 sup z∈C |f (z)|ε n (| max(ℓ + , λ YnY * n max ) -µ r )| -1 + |µ ℓ | -1 ).
This term converges to zero as n → ∞.

Proof of

Proposition 2.3.2 E x * M n x n n - TrM n n 2 ≤ K n (2.6.7)
has been proved in [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF]Eq. (3.2)]. Write

E y n M n y n - 1 n TrM n -a n M n a n 2 ≤K E 1 n x n M n x n - 1 n TrM n 2 + K n E|a * n M n x n | 2 + K n E|x * n M n a n | 2 .
The first term is with the order O(n -1 ) by (2.6.7). The second and the third terms are similar and we treat

1 n E|a * n M n x n | 2 : 1 n E|a * n M n x n | 2 = 1 n E(x * n M * n a n a * n M n x n ) ≤ K n E 1 n x * n M * n a n a * n M n x n - 1 n Tr(M * n )a j a * j M n + K n E 1 n TrM n a n a * n M n .
The first term in the above inequality is with the order O(n -1 ) by (2.6.7) and the second term is of the right order.

Proof of Lemma 2.3.3

From

E 1 n x * n M n x n - 1 n TrM n + 1 √ n a * n M n x n + 1 √ n x * n M n a n 4 ≤KE 1 n x * n M n x n - 1 n TrM n 4 + 1 n 2 E|a * n M n x n | 4 + 1 n 2 E|x * n M n a n | 4 ,
by [11, Eq. (3.

2)], we have

E 1 n x * n M n x n - 1 n TrM n 4 ≤ Kη 4 n n .
The rest can be treated as following :

1 n 2 E|x * n M n a n | 4 = 1 n 2 E|x * n M n a n a * n M * n x n | 2 ≤ 2 E 1 n x * n M n a n a * n M n x n - 1 n TrM n a n a * n M n 2 + E 1 n TrM n a n a * n M n 2 .
Thanks to Lemma 2.3.2, the first term is with the order O(n -2 ) and the second term is directly of the right order while 1 n 2 E|a * n M n x n | 4 can be treated similarly. Then we have

E 1 n x * n M n x n - 1 n TrM n + 1 √ n a * n M n x n + 1 √ n x * n M n a n 4 ≤ Kη 4 n n .
2.6.5 Proof of Lemma 2.3.5

We first prove (2.3.29). After the truncation step (cf. Section 2.3.2), for z ∈ {x

+ id; x ∈ [µ ℓ , µ r ]} E Q(z) p ≤ KE Q(z) < ∞. Take η ℓ ∈ (µ ℓ , 0) and η r ∈ (ℓ + , µ r ). For z ∈ {µ ℓ + iv; v ∈ [ε n /n, d]} ∪ {µ ℓ + iv; v ∈ [ε n /n, d]}, we get E Q j (z) p ≤ K + Kv -p P(λ Y j Y * j max ≥ η r or λ Y j Y * j min ≤ η ℓ ) ≤ K + Kn p ε -p n n -k ≤ K p , for sufficiently large k. This proves E Q j (z) p ≤ K p .

Now we will show

E|y * j Q q j (z)y j | p ≤ K pq . (2.6.8)
With the same method, (2.6.8) holds true for p = 1. For p ≥ 2,

E|y * j Q q j (z)y j | p ≤ K + Kv -qp E|y * j y j | p P(λ Y j Y * j max ≥ η r or λ Y j Y * j min ≤ η ℓ ) ≤ K + Kn qp ε -qp n η 2p-4 n E|y * j y j | 2 n -k ≤ K qp ,
for large enough k. This terminates the proof of (2.3.29). Following the same idea, we will prove (2.3.30). From Lemma 2.3.2, (2.3.30) holds true for z ∈ {x

+ id; x ∈ [µ ℓ , µ r ]}. For z ∈ {µ ℓ + iv; v ∈ [ε n /n, d]} ∪ {µ ℓ + iv; v ∈ [ε n /n, d]}, noticing that M j = Q k 1 j (z 1 )Q k 2 j (z 2 ) ≤ v -k 1 1 v -k 2 2 ,
we have then

E y * j M j y j -n -1 TrM j -a * j M j a j 2 ≤ K/n + Kv -k 1 1 v -k 2 2 P(λ YY * max ≥ η r or λ YY * min ≤ η ℓ ) ≤ K + Kn k 1 +k 2 ε -k 1 -k 2 n n -k ≤ K 2k 1 +2k 2 /n,
for k sufficiently high.

(2.3.31) can be shown in a similar way.

We now show that for z ∈ C n ,

E|γ j (z)| 2 ≤ K/n.
(2.6.9)

From [11, Eq (3.5)],

E|n -1 x * j Q j x j -n -1 ETrQ j | 2 ≤ K/n. Then E|γ j (z)| 2 = E|n -1 x * j Q j x j -n -1 ETrQ j + n -1/2 x * j Q j a j + n -1/2 a * j Q j x j | 2 ≤ 3E|n -1 x * j Q j x j -n -1 ETrQ j | 2 + 3 n E|x * j Q j a j | 2 + 3 n E|a * j Q j x j | 2 ≤ K/n.
where the last inequality follows from the fact that by (2.3.29),

max 1 n E|a * j Q j x j | 2 , 1 n E|x * j Q j a j | 2 ≤ K/n.
The same way also handles

E|γ j (z)| 4 ≤ Kη 4 n /n.
Since β j = 1y * j Qy j , same method yields that, for p ≥ 2,

E|β j | p = E|1 -y j Qy j | p ≤ K + Kv -p E|y j y * j | p P(λ YY * max ≥ η r or λ YY * min ≤ η ℓ ) ≤ K + Kn p ε -p n η 2p n n -k ≤ K p .
Since b j = β j + b j β j γ j along with (2.6.9), we have

|b j | ≤ |Eβ j + Eb j β j γ j | ≤ K 1 + K 2 |b j |n -1/2 ,
which implies

|b j (z)| ≤ K 1 1 -K 2 n -1/2 < ∞.
This terminates the proof of (2.3.33).

Proof of Lemma 2.4.6

From Equation (2.4.6), we have

a * E j Q 1 ME j Q 2 a -Ea * E j Q 1 ME j Q 2 a = j i=1 (E i -E i-1 )(a * E j Q 1 ME j Q 2 a) = j i=1 (E i -E i-1 )(a * E j [Q 1,i + z 1 q1,ii Q 1,i y i y * i Q 1,i ]M × E j [Q 2,i + z 2 q2,ii Q 2,j y i y * i Q 2,i ]a) = j i=1 z 2 (E i -E i-1 )[a * (E j Q 1,i )M(E j q2,ii Q 2,j y i y * i Q 2,i )a + j i=1 a * z 1 q1,ii Q 1,i y i y * i Q 1,i M(E j Q 2,i )a + j i=1 z 1 z 2 E j (q 1,ii q2,ii a * Q 1,i y i y * i Q 1,i )M(E j Q 2,i y i y * i Q 2,i a)] △ = X 1 + X 2 + Z.
(2.6.10)

The variance of a * E j Q 1 ME j Q 2 a is the sum the variance of these martingale increments. Recall that qii = bi + z qii bi γi ,

X 1 = z 2 j i=1 (E i -E i-1 )( b2,i y * i Q 2,i aa * (E j Q 1,i )MQ 2,i y i ) + z 2 2 j i=1 (E i -E i-1 )( b2,i q2,ii γ2,i y * i Q 2,i aa * (E j Q 1,i )MQ 2,i y i ) △ = X ′ 1 + X ′′ 1 . Let M i (z 1 , z 2 ) = Q 2,i aa * (E j Q 1,i )MQ 2,i . X ′ 1 satisfies : E|X ′ 1 | 2 = |z 2 | 2 j i=1 E E i bi r * i M i r i - 1 n TrM i + r * i M i a i + a i M i r i 2 .
Since M i is a rank one matrix, by Lemma 2.4.1,

j i=1 E|r * i M i r * i -TrM i /n| 2 ≤ K/n. Moreover, j i=1 E|r * i M i a i | 2 = 1 n j i=1 E(a * Q * 2,i Q 2,i a|a * (E j Q 1,i )MQ 2,i a i | 2 ) ≤ K n j i=1 E|a * (E j Q 1,i )MQ 2,i a i | 2 . Then |a * (E j Q 1,i )MQ 2,i a i | 2 ≤4(|a * (E j Q 1 )MQ 2 a i | 2 + |a * (E j Q 1,i -Q 1 )M(Q 2,i -Q 2 )a i | 2 + |a * (E j Q 1 )M(Q 2,i -Q 2 )a i | 2 + |a * (E j (Q 1,i -Q 1 ))MQ 2 a i | 2 ) △ =4(W i,1 + W i,2 + W i,3 + W i,4
).

Denote by

A 1:j = [a 1 , • • • , a j ],
we have :

j i=1 W i,1 = a * (E j Q 1 )MQ 2 A 1:j A * 1:j (E j Q * 1 )MQ * 2 a ≤ K.
Recalling Equation (2.4.7) and (2.4.8), recalling

1/β i = 1 + y * i Q i y i , we get W i,2 ≤ a * Q 2 y i y * i Q * 2 a i 1 -y * i Q 2 y i × a * E j Q 1 y i y * i Q 1 β 1,i M Q 2 y i y * i Q 2 1 -y * i Q 2 y i ME j Q 1 y i y * i Q 1 β 1,i a.
As (1 -

y * i Qy i ) -1 Qy i y * i Q = Q -Q i ≤ K and QY ≤ K, we have j i=1 EW i,2 ≤ j i=1 E |a * Q 1 y i | 2 |β 1,i | 2 . Noticing that β -1 i = -(z bi ) -1 + γi , by Lemma 2.
3.2 we obtain :

j i=1 EW i,2 ≤ 2Ea * Q 1 Y 1:j diag(-(z 1 b1,1 ) -1 , • • • , -(z 1 b1,j ) -1 )Y 1:j Q 1 a + K j i=1 E|γ 1 | 2 ≤ K.
The terms W i,3 and W i,4 can be handled similarly. Therefore we get j i=1

E|y * i M i a i | 2 ≤ K/n and j i=1 E|a * i M i y i | 2 ≤ K/n which implies that E|X 1 | 2 ≤ K/n. We now consider X 2 . Since E|X 2 | 2 ≤ 2|z 2 | 4 j i=1 E| b2,i q2,ii γ2,i y * i M i y i | 2 , we have j i=1 E| b2,i q2,ii γ2,i a * i M i a i | 2 ≤ K j i=1 EE (i) |γ 2,i a * i M i a i | 2 ≤ K n j i=1 E|a * i M i a i | 2 ≤ K n j i=1 E|a * i Q 2,i a i | 2 ≤ K n ,
where

E (i) = E[.|r 1 , • • • , r i-1 , r i+1 , • • • , r n ]. Moreover, by Lemma 2.3.3, j i=1 E| b2,i q2,ii γ2,i r * i M i a i | 2 ≤ K j i=1 E 1/2 |γ 2,i | 4 E 1/2 |r * i M i a i | 4 ≤ Kη 2 n √ n ,
and similarly for the terms in

a * i M i r i and r * i M i r i . We get that E|X 2 | 2 = o(n -1/2
). We now turn to the term Z of Equation (2.6.10). By developing y i = r i + a i , we have three terms to control,

Z 1 = j i=1 (E i -E i-1 )(E j q1,ii a * Q 1,i r i y * i Q 1,i )M(E j q2,ii Q 2,i r i a * i Q 2,i a) Z 2 = j i=1 (E i -E i-1 )(E j q1,ii a * Q 1,i r i y * i Q 1,i )M(E j q2,ii Q 2,i r i y * i Q 2,i a) Z 3 = j i=1 (E i -E i-1 )(E j q1,ii a * Q 1,i r i a * i Q 1,i )M(E j q2,ii Q 2,i r i a * i Q 2,i a).
The first term satisfies

E|Z 1 | 2 ≤ 2 j i=1 E|(E j q1,ii a * Q 1,i r i a * i Q 1,i )M(E j q2,ii Q 2,i r i a * i Q 2,i a)| 2 ≤ KE|(E j a * Q 1,i r i a * i Q 1,i )M(Q 2,i r i a * i Q 2,i a)| 2 = K j i=1 E|a * i Q 2,i a| 2 E (i) |(E j a * Q 1,i r i a * i Q 1,i )MQ 2,i r i | 2 ≤ K n j i=1 E|a * i Q 2,i a| 2 = O(n -1 ),
where the second inequality comes from

E|E j (X)E j (Y )| 2 = E|E j (XE i (Y ))| 2 ≤ E|XE j (Y )| 2 .
The terms Z 2 and Z 3 can be handled similarly. Hence

E|Z| 2 ≤ K n . Therefore, Var(a * (E j Q 1 )M(E j Q 2 )a) = o(1/ √ n). Var(Tr[(E j Q 1 )M(E j Q 2 )]) = o( √ n) can
α ℓ u * T 1 a ℓ E[a * ℓ Q 2 u] 1 + z 2 t2,ℓℓ a * ℓ T 2,ℓ a ℓ = j ℓ=1 -α ℓ u * T 1 a ℓ a * ℓ T 2 u z 2 t2,ℓℓ (1 + δ 2 ) + O(n -1/2 ).
Moreover, by Equation (2.4.6),

j ℓ=1 α ℓ u * T 1 a ℓ E[a * ℓ Q 2 u] 1 + z 2 t2,ℓℓ a * ℓ T 2,ℓ a ℓ -E[a * ℓ Q 2,ℓ u] = j ℓ=1 α ℓ u * T 1 a ℓ E a * ℓ Q 2,ℓ u 1 + z 2 q2,ℓℓ a * ℓ Q 2,ℓ y ℓ 1 + z 2 t2,ℓℓ a * ℓ T 2,ℓ a ℓ -1 + j ℓ=1 α ℓ u * T 1 a ℓ E[z 2 q2,ℓℓ a * ℓ Q 2,ℓ y ℓ r * ℓ Q 2,ℓ u] 1 + z 2 t2,ℓℓ a * ℓ T 2,ℓ a ℓ △ = ǫ 1 + ǫ 2 .
We have then

ǫ 1 = j ℓ=1 α ℓ u * T 1 a ℓ E[a * ℓ Q 2,ℓ uµ 2,ℓ ] where E|µ 2,ℓ | 2 ≤ Kn -1 by (2.4.18) again. It follows that |ǫ 1 | ≤ ( j ℓ=1 α 2 ℓ |u * T 1 a ℓ | 2 E|µ 2,ℓ | 2 ) 1/2 ( j ℓ=1 E|a * ℓ Q 2,ℓ u| 2 ) 1/2 ≤ K √ n
by (2.4.12) in Theorem 2.4.1. By writing

ǫ 2 = j ℓ=1 α ℓ u * T 1 a ℓ E[z 2 (q 2,ℓℓ a * ℓ Q 2,ℓ y ℓ -E[q 2,ℓℓ a * ℓ Q 2,ℓ y ℓ ])r * ℓ Q 2,ℓ u] 1 + z 2 t2,ℓℓ a * ℓ T 2,ℓ a ℓ
and proceeding similarly to ǫ 1 , we obtain |ǫ 2 | ≤ K/ √ n which completes the proof.

Proof of Lemma 2.4.8

For a matrix A, define

Ã1:j = [ã 1 , • • • , ãj , 0, • • • ],
where a j it the j-th column of A. We will show that

η j + ω j = z 1 z 2 n
Tr T1:j T * 1:j .

(2.6.11)

In particular, j = n implies [START_REF] Anderson | An introduction to random matrices[END_REF].

From the definition of ω j , we have

ω j = 1 n (1 + δ 1 ) 2 (1 + δ 2 ) 2 TrA * 1:j T 1 A 1:j A * 1:j T 2 A 1:j - 1 n j k=1 a * k T 1 a k a * k T 2 a k (1 + δ 1 ) 2 (1 + δ 2 ) 2 .
From (2.4.9) and (2.4.5),

(1 + δ) -2 a * k T k a k = (1 + δ) -1 + z tkk . Hence 1 n j k=1 a * k T 1 a k a * k T 2 a k (1 + δ 1 ) 2 (1 + δ 2 ) 2 = 1 n j k=1 z 1 z 2 t1,kk t2,kk + 1 n j k=1 z 1 t1,kk 1 + δ 2 + 1 n j k=1 a * k T 2 a k (1 + δ 1 )(1 + δ 2 ) 2 .
By -z T1:j = (1 + δ) -1 I 1:j -(1 + δ) -2 A * 1:j TA 1:j , we have

η j + ω j = - 1 n(1 + δ 2 )
Tr T1:j (z 1 ) -

1 n(1 + δ 1 )(1 + δ 2 ) 2 TrA * 1:j T 2 A 1:j + 1 n(1 + δ 1 ) 2 (1 + δ 2 ) 2 TrA * 1:j T 1 A 1:j A * 1:j T 2 A 1:j = z 1 z 2 n
Tr[ T1:j (z 1 ) T1:j (z 2 )]. Now we will prove (2). From (3.1.1),

T(z) 1 + δ = (-z(1 + δ(z))(1 + δ(z))I N + AA * ) -1
(2.6.12)

T(z) 1 + δ = (-z(1 + δ(z))(1 + δ(z))I n + A * A) -1 .
(2.6.13) By (2.6.12)-(2.6.13) and taking trace, we have

δ -δ = n -N n 1 z . (2.6.14)
Using the resolvent formula, we get

T 1 1 + δ 1 - T 2 1 + δ 2 = z 1 (1 + δ1 )(1 + δ 1 ) -z 2 (1 + δ2 )(1 + δ 2 ) (1 + δ 1 )(1 + δ 2 ) T 1 T 2 .
(2.6.15)

Taking trace on both sides of the precedent equality, we have

γ = 1 n TrT 1 T 2 = δ 1 (1 + δ 2 ) -δ 2 (1 + δ 1 ) z 1 (1 + δ1 )(1 + δ 1 ) -z 2 (1 + δ2 )(1 + δ 2 ) = δ 1 -δ 2 z 1 (1 + δ1 )(1 + δ 1 ) -z 2 (1 + δ2 )(1 + δ 2 )
.

In the same way, we get that

γ = δ1 -δ2 z 1 (1 + δ1 )(1 + δ 1 ) -z 2 (1 + δ2 )(1 + δ 2
) .

We will express 1 n TrT 1 AA * T 2 with the help of δ 1 , δ 2 . We obtain,

1 n TrT 1 AA * T 2 = 1 n TrT 1 (AA * -z 1 (1 + δ1 )(1 + δ 1 ) + z 1 (1 + δ1 )(1 + δ 1 ))T 2 = (1 + δ 1 )δ 2 + z 1 (1 + δ1 )(1 + δ 1 ) n TrT 1 T 2 = (1 + δ 1 )(1 + δ 2 ) z 1 (1 + δ1 )δ 1 -z 2 (1 + δ2 )δ 2 z 1 (1 + δ1 )(1 + δ 1 ) -z 2 (1 + δ2 )(1 + δ 2 )
.

Then

1 - 1 n 1 1 + δ 1 1 1 + δ 2 TrT 1 AA * T 2 = z 1 (1 + δ1 ) -z 2 (1 + δ2 ) z 1 (1 + δ1 )(1 + δ 1 ) -z 2 (1 + δ2 )(1 + δ 2 )
.

(2.6.16)

Replace the expressions of γ, γ and 1 n TrT 1 AA * T 2 in ∆ n , we get :

∆ n = (z 1 (1 + δ1 ) -z 2 (1 + δ2 )) 2 -z 1 z 2 (δ 1 -δ 2 )( δ1 -δ2 )) (z 1 (1 + δ1 )(1 + δ 1 ) -z 2 (1 + δ2 )(1 + δ 2 )) 2 ,
and

(z 1 (1 + δ1 ) -z 2 (1 + δ2 )) 2 -z 1 z 2 (δ 1 -δ 2 )( δ1 -δ2 ) = z 2 1 (1 + δ1 ) 2 + z 2 2 (1 + δ2 ) 2 -2z 1 z 2 (1 + δ1 )(1 + δ2 ) -z 1 z 2 ( δ1 + n -N nz 1 -δ2 - n -N nz 2 )( δ1 -δ2 ) = z 2 1 (1 + δ1 ) 2 + z 2 2 (1 + δ2 ) 2 -z 1 z 2 -z 1 z 2 (( δ1 + 1) 2 + ( δ2 + 1) 2 ) + n -N n (z 1 -z 2 )( δ1 -δ2 ) = (z 1 -z 2 ) z 1 (1 + δ1 ) 2 -z 2 (1 + δ2 ) 2 + n -N n ( δ1 -δ2 ) = (s n (z 1 ) -s n (z 2 ))(z 1 -z 2 )
where the last equality follows from Equation (2.6.14). On the other hand,

(z 1 (1 + δ1 )(1 + δ 1 ) -z 2 (1 + δ2 )(1 + δ 2 )) 2 = (z 1 (1 + δ1 )(1 + δ1 + n -N nz 1 ) -z 2 (1 + δ2 )(1 + δ 2 )) 2 = (s n (z 1 ) -s n (z 2 )) 2 .
Thus,

1 ∆ n (z 1 , z 2 ) = s n (z 1 ) -s n (z 2 ) z 1 -z 2 .
The last task is to show that for

z 1 , z 2 ∈ C + , lim inf n inf j∈{1,••• ,n} |∆ j (z 1 , z 2 )| > 0.
We first prove that for all j ∈ {1, • • • , n} and z ∈ C + ,

Im[z(1 + δ)(1 + δ)] > 0,
(2.6.17)

ν j (z, z) < 1, (2.6.18
)

lim inf n inf j∈{1,••• ,n} ∆ j (z, z) > 0. (2.6.19) From (2.1.4), δ 1 + δ = 1 n n j=1 1 a i -z(1 + δ)(1 + δ)
where a i ∈ R + are eigenvalues of AA * .

As Im(δ) > 0, we get

Im δ 1 + δ = Im 1 - 1 1 + δ > 0, which leads to Im[z(1 + δ)(1 + δ)] > 0.
(2.6.20)

Recall the definition of ν j in (2.4.51) and by (2.6.16),

1 -ν n (z, z) = z(1 + δ(z)) -z(1 + δ(z)) z(1 + δ(z))(1 + δ(z)) -z(1 + δ(z))(1 + δ(z)) > 0,
hence (2.6.18). From (2) in Lemma 2.4.8,

0 < 1 ∆ n (z, z) = s n (z) -s n (z) z - z ≤ K.
This proves lim inf n ∆ n (z, z) > 0.

(2.6.21)

As ν j (z, z) is increasing to ν n (z, z), we get

ν j (z, z) < 1.
From the definition of ∆ n and (1) in Lemma 2.4.8,

∆ n (z, z) = (1 -ν n (z, z)) 2 -γ(z, z)(η n (z, z) + ω n (z, z)).
ν j (z, z) (resp. η j (z, z), ω j (z, z)) is increasing to ν n (resp. η n (z, z), ω n (z, z)), we have

lim inf n inf j∈{1,••• ,n} ∆ j (z, z) ≥ lim inf n ∆ n (z, z) > 0.
Hence (2.6. [START_REF] Cabanal-Duvillard | Fluctuations de la loi empirique de grandes matrices aléatoires[END_REF]). With these tools, we have

|ν j (z 1 , z 2 )| ≤ ν j (z 1 , z1 ) ν j (z 2 , z2 ) < 1.
(2.6.22)

We end up the proof by showing that for all z 1 , z 2 ∈ C + ,

|∆ j (z 1 , z 2 )| ≥ (∆ j (z 1 , z1 )∆ j (z 2 , z2 )) 1/2 . (2.6.23)
In particular, this directly implies

lim inf n inf j |∆ j (z 1 , z 2 )| ≥ lim inf n inf j ∆ j (z 1 , z1 ) lim inf n inf j ∆ j (z 2 , z2 ) 1/2 > 0.
From (2.6.11), we get

|∆ j (z 1 , z 2 )| ≥ (1 -|ν j (z 1 , z 2 )|) 2 - |z 1 z 2 | n |γ||Tr T1:j (z 1 ) T * 1:j (z 2 )| ≥ (1 -ν 1/2 j (z 1 , z1 )ν 1/2 j (z 2 , z2 )) 2 - |z 1 z 2 | n γ 1/2 (z 1 , z1 )γ 1/2 (z 2 , z2 )× (Tr T1:j (z 1 ) T * 1:j (z 1 )) 1/2 (Tr T1:j (z 2 ) T * 1:j (z 2 )) 1/2 .
(2.6.24)

By a direct calculation, we have :

for 0 ≤ a, b ≤ 1, (1 - √ ab) 2 ≥ (1 -a)(1 -b).
(2.6.25)

We first show that the right side of (2.6.24) is positive. As (1-ν j (z, z)) 2 ≥ |z| 2 n γ j (z, z)TrT 1:j (z)T * 1:j (z), we have

|z 1 z 2 | n γ 1/2 (z 1 , z1 )γ 1/2 (z 2 , z2 )(Tr T1:j (z 1 ) T * 1:j (z 1 )) 1/2 (Tr T1:j (z 2 ) T * 1:j (z 2 )) 1/2 ≤ (1 -ν j (z 1 , z1 ))(1 -ν j (z 2 , z2 )) ≤ (1 -ν 1/2 j (z 1 , z1 )ν 1/2 j (z 2 , z2 )) 2 .
Then we will show that the right hand side of (2.6.24) is greater than

(∆ j (z 1 , z1 )∆ j (z 1 , z1 )) 1/2 .
Denote for i = 1, 2,

a i = ν j (z i , zi ), b i = |z i | 2 n γ(z i , zi )(Tr T1:j (z i ) T * 1:j (z i )).
We will show that

[(1 - √ a 1 √ a 2 ) 2 -b 1/2 1 b 1/2 2 ] 2 ≥ ((1 -a 1 ) 2 -b 1 )((1 -a 2 ) 2 -b 2 ).
It suffices to show that

(1 - √ a 1 √ a 2 ) 2 ((1 - √ a 1 √ a 2 ) 2 -2b 1/2 1 b 1/2 2 ) ≥ (1 -a 1 ) 2 (1 -a 2 ) 2 -b 1 (1 -a 2 ) 2 -b 2 (1 -a 1 ) 2 . By (1 - √ a 1 √ a 2 ) 2 ≥ (1 -a 1 )(1 -a 2 ), it suffices to show that (1 - √ a 1 √ a 2 ) 2 -2b 1/2 1 b 1/2 2 ≥ (1 -a 1 )(1 -a 2 ) -b 1 1 -a 2 1 -a 1 -b 2 1 -a 1 1 -a 2 ,
which is immediate.

2.6.9 Proof of Lemma 2.4.10

Following the same proof as Lemma 2.4.8, noticing that |ϑ| ≤ 1, we have

|∆ ϑ j (z 1 , z 2 )| ≥ (1 -|ϑ||ν † j |)(1 -| θ||ν † j |) - |ϑ 2 z 1 z 2 | n γ † |Tr T1:j (z 1 ) TT 1:j (z 2 )| ≥ (1 -ν 1/2 j (z 1 , z1 )ν 1/2 j (z 2 , z2 )) 2 - |z 1 z 2 | n γ 1/2 (z 1 , z1 )γ 1/2 (z 2 , z2 )×
(Tr T1:j (z 1 ) T * 1:j (z 1 )) 1/2 (Tr T1:j (z 2 ) T * 1:j (z 2 )) 1/2 . This is the right hand side of (2.6.24). Thus,

lim inf n inf j |∆ ϑ j (z 1 , z 2 )| > 0.
Chapitre 3

Fluctuations for non-analytic functionals of linear spectral statistics

The content below is inspired from [START_REF] Najim | Fluctuations for linear spectral statistics of large random matrices :from analytic to non-analytic functionals[END_REF]. Consider an N × n matrix Y n with real random entries. Of particular interest is the study of the fluctuations of linear spectral statistics of large random covariance matrices Y n Y T n :

Tr f (Y n Y T n ) = N i=1 f (λ i ), (λ i ) eigenvalues of Y n Y T n ,
as the dimensions of matrix Y n go to infinity at the same pace :

N, n → ∞ and 0 < lim inf N n ≤ lim sup N n < ∞ , (3.0.1)
(a condition that will be simply referred as N, n → ∞ in the sequel). This subject has a rich history (see for instance [2,[START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF][START_REF] Lytova | Central Limit Theorem for linear eigenvalue statistics of random matrices with independent entries[END_REF][START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF] and the references therein) among which we single out the contributions [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF][START_REF] Pan | Central limit theorem for signal-to-interference ratio of reduced rank linear receiver[END_REF][START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF][START_REF] Najim | Fluctuations for linear spectral statistics of large random informationplusnoise type matrices[END_REF].

General principles and main results

The main idea is in Introduction 1.2. If ϕ is a random variable, denote by

• ϕ = ϕ -Eϕ.

Two lemmas

Lemma 3.1.1 Let (ϕ n (t), t ∈ R) n∈N and (ψ(t), t ∈ R) be complex-valued continuous random processes. Assume that ψ is centered and that : (i) The following convergence in distribution holds true :

∀t i ∈ R, 1 ≤ i ≤ d, • ϕ n (t 1 ), • • • , • ϕ n (t d ) D ---→ n→∞ (ψ(t 1 ), • • • , ψ(t d )).
(ii) For all T > 0, ϕ n (t) is tight on [-T, T ].

(iii) The following estimates hold true

(Varϕ n (t)) 1/2 ≤ K(t) and (Varψ(t)) 1/2 ≤ K(t) ,
where t → K(t) does not depend on n and is such that

R |K(t)| dt < ∞. Then, R • ϕ n (t) dt D ---→ n→∞ R ψ(t) dt Lemma 3.1.
1 should be compared with [50, Th. 4.28] and the proof is in Appendix 3.5.1. In the context of large random matrices, an interesting phenomenon occurs [START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF][START_REF] Najim | Fluctuations for linear spectral statistics of large random informationplusnoise type matrices[END_REF], where sometimes there is not a limiting process ψ(t) but where there exists instead a sequence of processes (ψ n ) whose distribution gets closer to the distribution of the random process of interest

( • ϕ n ).
The following lemma formalizes this. Lemma 3.1.2 Recall the definition of Lévy-Prohorov distance in Section 2.2 and let (ϕ n (t), t ∈ R) n∈N and (ψ n (t), t ∈ R) n∈N be complex-valued continuous random processes. Assume that all the ψ n 's are centered and that :

(i) The following convergence in distribution holds true :

∀t i ∈ R, 1 ≤ i ≤ d, d LP ( • ϕ n (t 1 ), • • • , • ϕ n (t d )), (ψ n (t 1 ), • • • , ψ n (t d )) ---→ n→∞ 0 (ii) For all T > 0, ϕ n (t)
d LP R • ϕ n (t) dt , R ψ n (t) dt ---→ n→∞ 0.
The proof is the same as Lemma 3.1.1 and is omitted here.

Fluctuations of the linear statistics

We consider the model

Y n = 1 √ n R 1/2 n X n .
For the reason of readability, we often drop subscripts n if there is no confusion.

Assumption 3.1.1 The random variables (x ij ) 1≤i≤N,1≤j≤n are real, independent and identically distributed (i.i.d.). They satisfy

Ex 11 = 0, E|x 11 | 2 = 1 and µ 4 = E|x 11 | 4 < ∞. Assumption 3.1.2 When N, n → ∞, 0 < lim inf N n ≤ lim sup N n < ∞,
which will be simply denoted by N, n → ∞ in the sequel.

Assumption 3.1.3 The family of N × N symmetric, non-negative definite matrix (R n ) is bounded for the spectral norm :

r max = sup n≥1 R n < ∞. Denote U n (t) = e itYnY T n , u(t) = TrU n (t), ůn (t) = u n (t) -Eu n (t), V n (t) = Var[u(t)].
The purpose of the chapter is, under the precedent conditions to study the fluctuations of the linear statistics of the eigenvalues of 1 n R 1/2 XX T R 1/2 for non-analytic functionals. For a sufficiently regular function f (for example : of class C 4 ), We will study the fluctuations of linear spectral analysis

n j=1 f (λ j ) = Trf (Y n Y T n ),
where λ j are eigenvalues of Y n Y T n . This issue was initially studied by Jonsson [START_REF] Jonsson | Some limit theorems for the eigenvalues of a sample covariance matrix[END_REF] for the moments of the eigenvalues of sample covariance matrix. The linear spectral analysis is first discovered by Bai and Silverstein in [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF]. In [START_REF] Hachem | A CLT for informationtheoretic statistics of non-centered Gram random matrices[END_REF], Hachem et al. establishes a CLT for mutual information log det(Y n Y T n + ρI N ). Recently, Najim [START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF] made a fully study of the model for analytic functionals and Najim and Yao [START_REF] Najim | Fluctuations for linear spectral statistics of large random informationplusnoise type matrices[END_REF] generalized the result for information-plus-noise model. However, all studies are based on the assumption that f is an analytic function on a certain open region.

Some more notations

The Stieljes transform associated to a random matrix

Y n Y T n (resp. Y T n Y n ) is defined by m n (z) = 1 N TrQ n (z) (resp. mn (z) = 1 n Tr Qn (z)) where Q n (z) = (Y n Y T n -zI n ) -1 (resp. Qn (z) = (Y T n Y n -zI n ) -1 ) is the resolvent associated to Y n Y T n (resp.Y T n Y n ).
We define the matrix T n (z) by

T n (z) = (-z(I N + t(z)R n )) -1 . (3.1.1)
For z ∈ C\R + , the system

   t n (z) = 1 N Tr(-z(I N + t(z)R n )) -1 , tn (z) = - 1 z(1+ 1 n TrRnTn) (3.1.2)
admits a unique solution (t n , tn ) from C + to C + . T n approximate the resolvents Q n (z) in the sense that for z ∈ C\R + ,

1 n Tr(Q n (z) -T n (z))
a.s.

---→ n→∞ 0.

It has also been proved in [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF] that t n (resp. tn ) is the Stieljes transform of a probability distribution which is denoted by F n (resp. Fn ). By Silverstein and Choi [START_REF] Silverstein | Analysis of the limiting spectral distribution of large dimensional random matrices[END_REF], the limit of tn exists as z ∈ C + approaches the real axis :

lim z∈C + →x tn (z) = tn (x), for x ∈ R * .
In the sequel, we denote S n the support associated with the measure Fn of deterministic equivalent tn .

Denote

G n = N (F YnY T n -F n ).
The central object is the study of the random variable

N (f ) △ = R f (x)G(dx). (3.1.3)
In particular, we will cut the random variable into random part N 1 (f ) and deterministic part

N 2 (f ) : N (f ) △ = N 1 (f ) + N 2 (f )
where

N 1 (f ) = N R f (x)F YnY T n (dx) -E R f (x)F YnY T n (dx) (3.1.4) N 2 (f ) = N E R f (x)F YnY T n (dx) - R f (x)F n (dx) . (3.1.5)
For s 1 , s 2 ∈ R, we denote

K n (s 1 , s 2 ) = s 1 s 2 tn (s 1 ) tn (s 2 ) n N i=1 [R 1/2 n T n (s 1 )R 1/2 n ] ii [R 1/2 n T n (s 2 )R 1/2 n ] ii , (3.1.6)
K † n (s 1 , s 2 ) = s 1 s 2 tn (s 1 ) tn (s 2 ) n N i=1 [R 1/2 n T n (s 1 )R 1/2 n ] ii [R 1/2 n T n (s 2 )R 1/2 n ] ii . (3.1.7) Notice that if s 1 / ∈ S n or s 2 / ∈ S n , K n (s 1 , s 2 ) = K † n (s 1 , s 2 ) = 0.
We will show the following theorem. and define the Gaussian vector

Z 1 n (f ) = (Z 1 n (f 1 ), • • • , Z 1 n (f k ))
with zero mean and the covariance

Cov[Z 1 n (f i ), Z 1 n (f j )] = 1 2π 2 Sn Sn f ′ i (s 1 )f ′ j (s 2 ) log 1 + 4Im[ tn (s 1 )]Im[ tn (s 2 )] | tn (s 1 ) -tn (s 2 )| 2 ds 1 ds 2 - κ 4 2π 2 Sn Sn f ′ i (s 1 )f ′ j (s 2 )Re(K n (s 1 , s 2 ) -K † n (s 1 , s 2 ))ds 1 ds 2 , (3.1.9)
where κ 4 = µ 4 -3 is the fourth cumulant of X ij . Then

d LP (N 1 n (f ), Z 1 n (f )) -----→ N,n→∞ 0

Expression of the bias

In this section, we will express the bias N 2 (f ) defined in (3.1.5). Define

B 1,n (z) △ = - z 2 t3 n 1 n TrR 2 n T 3 n (1 -z 2 t2 n 1 n TrR 2 n T 2 n ) 2 (3.1.10) B 2,n (z) △ = - z 3 t3 n 1 n N i=1 [R 1/2 n T n R 1/2 n ] ii [R 1/2 n T 2 n R 1/2 n ] ii 1 -z 2 t2 n 1 n TrR 2 n T 2 n , ( 3
(1 + |t| 4 )| f (t)|dt < ∞.
Recall the definition of

N 2 n (f ) = (N 2 n (f 1 ), • • • , N 2 n (f k )) where N 2 n (f i ) is defined in (3.1.5) and define the vector Z 2 n (f ) = (Z 2 n (f 1 ), • • • , Z 2 n (f k ))
where

Z 2 n (f k ) = 1 π Sn f k (t)Im[B n (t)]dt, and B n (t) = B 1,n (t) + κ 4 B 2,n (t). Then N 2 n (f ) -Z 2 n (f ) -----→ N,n→∞ 0.

Estimation of the variance

The key point in the study of the fluctuations is an estimate of Var[u n (t)]. We shall estimate the variance in more general cases. Consider the model

Y n = 1 √ n R 1/2 n X n + A n ,
where A n is a sequence of deterministic matrices with bounded spectral measure and R n , X n satisfy Assumptions 3.1.1, 3.1.2, 3.1.3. We first need a truncated version of the entries.

Truncation of the random variables

As suggested in [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF], Najim [START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF], Najim and Yao [START_REF] Najim | Fluctuations for linear spectral statistics of large random informationplusnoise type matrices[END_REF], the first step consists in the truncation of the random variables, i.e., the replacement of the entries X ij satisfying that there exists a sequence

η n → 0 such that η n n 1/5 → ∞, |X ij | < η n √ n.
More precisely, by Assumption 3.1.1, we may select η n ↓ 0 such that :

1 η 4 n E|X 11 | 4 I(|X 11 | ≥ η n √ n) → 0, n → ∞. Let Ỹn = 1 √ n R 1/2 n Xn + A n with Xn , an N × n matrix with the entries Xij = ( Xij -E Xij )/σ ij where Xij = X ij I |X ij |≤ηn
√ n and σ ij = Var Xij . We have then

P(Y n Y T n = Ỹn ỸT n ) ---→ n→∞ 0,
and max |E Xij -EX ij |, E| Xij | 2 -1, E| Xij | 4 -E|X ij | 4 ---→ n→∞ 0.
Moreover, for future use, for any function f of class C 4 ,

P f (λ)dF YnY T n (dλ) -f (λ)dF Ỹn ỸT n (dλ) > ǫ -----→ N,n→∞ 0.
Hence it is sufficient to consider the truncated, centralized and renormalized variables. This will be assumed throughout the chapter. Then we have the estimate of the variance.

Theorem 3.1.3 Under Assumptions 3.1.1, 3.1.2, 3.1.3, let V n (t) = Var[u n (t)], we have V n (t) ≤ C 0 (µ 4 )(1 + |t| 4 ) 2 ,
where C 0 is a constant which only depends on µ 4 .

The idea behind this theorem is Fourier transform. For an integrable function f , we have :

For x ∈ R, f (t) = 1 2π R f (x)e -ixt dx
and by the inverse formula, we have

f (x) = R f (t)e ixt dt.
Given a function f , the linear statistics of the eigenvalue of YY T is

N (f ) = Trf YY T = N i=1 f (λ i ),
where the λ i 's are the eigenvalues of matrix YY T . Applying the precedent theorem, one has immediately that

Var(N (f )) ≤ C 0 (µ 4 ) (1 + |t| 4 )| f (t)|dt 2 .
(3.1.12)

Such an inequality has proved to be useful to establish CLTs for the linear statistics of large random covariance matrices, (cf. [START_REF] Lytova | Central Limit Theorem for linear eigenvalue statistics of random matrices with independent entries[END_REF][START_REF] Pastur | Eigenvalue distribution of large random matrices[END_REF]) and should be compared to logarithmic sobolev inequalities(LSI) and Talagrand concentration inequalities.

In the case of LSI, the constraint over f is lighter (f only needs to be Lipschitz with bounded Lipschitz norm -a fact that is induced by the condition that the right hand side of (3.1.12) should be bounded,) but the condition over the entries is much stronger as their distribution needs to satisfy a LSI, which essentially induces subgaussiannity.

In the case of Talagrand concentration inequalities, the regularity constraint over f is the same as LSI but there is a strong structure constraint as f needs to be convex as well (or to have a limited number of inflection points).

The chapter is organized as below. In Section 3.2, the proof of Theorem 3.1.3 is given for

Y n = 1 √ n R 1/2
n X n + A n in real case. When A n = 0, Section 3.3 gives the proof of Theorem 3.1.1 by means of Lemma 3.1.2. The expression of real integrals of the variance is also proposed. The computation of the bias in this case is in Section 3.4. Some technical proofs are in Appendices.

Proof of Theorem 3.1.3 : Estimation of the variance

Some preparations

More notations

We denote Xij , the N ×n random matrix with i.i.d. real standard gaussian entries. The terms related to Xij will be denoted as

Ŷn = 1 n R 1/2 n Xn + A n , Û(t) = e it Ŷn ŶT n , û(t) = Tr Û(t).
Denote by µ ℓ the moments EX ℓ ij and κ ℓ the associated cumulants. The technique proposed by Lytova and Pastur [START_REF] Lytova | Central Limit Theorem for linear eigenvalue statistics of random matrices with independent entries[END_REF] consists in considering the interpolating matrix

X s = √ sX + √ 1 -s X
and the related terms will be denoted by

Y(s) = 1 √ n R 1/2
n X s + A n or Y s when there is no confusion. Other related terms will be denoted by U(t, s) and u(t, s). Associated to these three matrices are the following quantities :

X X X s Y = 1 √ n R 1/2 X + A Ŷ = 1 √ n R 1/2 X + A Y s = 1 √ n R 1/2 X s + A U(t) = e itYY T Û(t) = e it Ŷ ŶT U(t, s) = e itYsY T s u(t) = TrU(t) û = Tr Û(t) u(t, s) = TrU(t, s)
The idea is to show Theorem 3.1.3 under Gaussian case. Then matrix X s will naturally appear when considering the difference

α(t) = u(t) -û(t) = 1 0 d ds u(t, s)ds
which is at the heart of the interpolation procedure.

Differentiation formulas

Differentiating matrix functionals with respect to matrices' entries will be important in the sequel. This can be done with the help of Duhamel's formula which can be shown by a simple integration by parts : Proposition 3.2.1 (Duhamel Formula) Let M 1 , M 2 be two n × n real symmetric or Hermitian matrices and t ∈ R. Then we have e iM 2 t = e iM 1 t + i t 0 e iM 1 (t-s) (M 2 -M 1 )e iM 2 s ds.

To describe the differential, we introduce the convolution notations. If f and g are two functions of t,

f * g(t) = t 0 f (t -t 1 )g(t 1 )dt 1 . Lemma 3.2.1 First order derivatives ∂ ∂Y jk [U (t)] mn = i([Y T U (t)] km * [U (t)] jn + [U (t)] mj * [Y T U (t)] kn ), (3.2.1)
∂ ∂Y jk [Y T U (t)] kj = [U (t)] jj + i[Y T U (t)Y ] kk * [U (t)] jj +i[Y T U (t)] kj * [Y T U (t)] kj , (3.2.2) ∂ ∂Y jk u(t) = 2it[Y T U (t)] kj , (3.2.3) ∂ ∂Y jk [Y T U (t)Y ] kk = 2[Y T U (t)] kj + 2i[Y T U (t)Y ] kk * [Y T U (t)] kj . (3.2.4)
Second order derivatives

∂ 2 ∂(Y jk ) 2 [Y T U (t)] kj = 6i[U (t)] jj * [Y T U (t)] kj -6[Y T U (t)Y ] kk * [Y T U (t)] kj * [U (t)] jj -2[Y T U (t)] kj * [Y T U (t)] kj * [Y T U (t)] kj , (3.2.5) ∂ 2 ∂(Y jk ) 2 u(t) = 2it([U (t)] jj + i[Y T U (t)Y ] kk * [U (t)] jj + [U (t)Y ] jk * [U (t)Y ] jk ). (3.2.6)
The proof is in Appendix 3.5.2.

In general case, we introduce also the following differentiation operator :

D αβ = ∂ ∂X αβ , and
D αβ (s) = ∂ ∂[X s ] αβ with the obvious iterations D ℓ αβ = ∂ ℓ ∂(X αβ ) ℓ . Since Y = 1 √ n R 1/2 X + A,
we obtain that

D jk [R 1/2 U (t)R 1/2 ] mn = i √ n ([R 1/2 U (t)Y ] mk * [R 1/2 U (t)R 1/2 ] jn +[R 1/2 U (t)R 1/2 ] mj * [Y T U (t)R 1/2 ] kn ), (3.2.7)
D jk [Y T U (t)R 1/2 ] kj = 1 √ n [R 1/2 U (t)R 1/2 ] jj + i √ n [Y T U (t)Y ] kk * [R 1/2 U (t)R 1/2 ] jj + i √ n [Y T U (t)R 1/2 ] kj * [Y T U (t)R 1/2 ] kj , (3.2.8)
D jk u(t) = 2it √ n [Y T U (t)R 1/2 ] kj , (3.2.9)
D jk [Y T U (t)Y ] kk = 2 √ n [Y T U (t)R 1/2 ] jk + 2i √ n [Y T U (t)Y ] kk * [Y T U (t)R 1/2 ] kj .
(3.2.10)

and second order derivatives

D 2 jk [Y T U (t)R 1/2 ] kj = 6i n [R 1/2 U (t)R 1/2 ] jj * [Y T U (t)R 1/2 ] kj (3.2.11) - 6 n [Y T U (t)Y ] kk * [Y T U (t)R 1/2 ] kj * [R 1/2 U (t)R 1/2 ] jj - 2 n [Y T U (t)R 1/2 ] kj * [Y T U (t)R 1/2 ] kj * [Y T U (t)R 1/2 ] kj , D 2 jk u(t) = 2it n ([R 1/2 U (t)R 1/2 ] jj + i[Y T U (t)Y ] kk * [R 1/2 U (t)R 1/2 ] jj +[Y T U (t)R 1/2 ] jk * [Y T U (t)R 1/2 ] jk ). (3.2.12)
Of course we may extend the previous formulas for

Djk = ∂ ∂ Xjk and D jk (s) = ∂ ∂[X s ] jk .
There are some bridges between these notations. Recall that X s = √ sX + √ 1s X. If Φ = Φ(X s ), then Φ can be considered as a function of X or X as well and we may use the notational shortcut Φ = Φ(X) (resp. Φ = Φ( X)). In this regard, we may use the convenient (but somehow hazardous) notation :

D jk Φ = √ sD jk (s)Φ (3.2.13) Djk Φ = √ 1 -sD jk (s)Φ.
In the different terms, the function Φ is respectively assumed to be a function of X s , then of Y, then of Ŷ ; the parameter of Φ being indicated by the differentiation operator.

Remark 3.2.1 Beware that if Φ is not a function of Y s , then (3.2.13) may not hold any more. Take for instance the simple example :

Φ = u(t) -û(t).
According to (3.2.9) :

D jk Φ = 2it √ n [Y T U (t)R 1/2 ] kj while Djk Φ = -2it √ n [ Ŷ T Û (t)R 1/2 ] kj .
By iterating the derivatives, although the expression of D ℓ jk [Y T U (t)R 1/2 ] kj might be lengthy, only three terms appear frequently :

[Y T U (t)R 1/2 ] kj , [R 1/2 U (t)R 1/2
] jj and [Y T U (t)Y ] kk . This is important to understand the technique.

Generalized integration by part formula

We recall the generalized integration by part formula, as exposed in Pastur and Shcherbina in [START_REF] Pastur | Eigenvalue distribution of large random matrices[END_REF]Proposition 18.1.4]. Proposition 3.2.2 Let ξ be a random variable such that E{|ξ| p+2 } < ∞ for a certain nonnegative integer p. Then for any function Φ : R → C of the class C p+1 with bounded derivatives Φ (ℓ) , ℓ = 1, ..., p + 1, we have

E{ξΦ(ξ)} = p ℓ=0 κ ℓ+1 ℓ! E{Φ (ℓ) (ξ)} + ε p ,
where the remainder term ǫ p admits the bound

|ε p | ≤ C p E{|ξ| p+2 } sup t∈R |Φ (p+1) (t)|, C p ≤ 1 + (3 + 2p) p+2 (p +
|Φ (p+1) (t)|.
In order to see this, consider the function

Φ =          Φ(t) if t ∈ [a, b], p+1 k=0 Φ (k) (a) (t-a) k k! if t < a, p+1 k=0 Φ (k) (b) (t-b) k k! if t > b.
Then Φ is of the class C (p+1) , the following equalities hold true :

E{ξ Φ(ξ)} = E{ξΦ(ξ)}, E Φ(k) (ξ) = EΦ (k) (ξ) and sup t∈R | Φ(p+1) (t)| = sup t∈[a,b] | Φ(p+1) (t)|.
Hence the sharper estimate.

Strategy of the proof

The general strategy of proof has been exposed in Lytova and Pastur [START_REF] Lytova | Central Limit Theorem for linear eigenvalue statistics of random matrices with independent entries[END_REF]. The main idea consists in interpolating between the linear statistics u(t) = Tre itYY T (based on truncated random variables) and the Gaussian linear statistics û(t) = Tre it Ŷ ŶT . In the Gaussian case, a straightforward application of Poincaré-Nash inequality [START_REF] Hachem | A new approach for capacity analysis of large dimensional multi-antenna channels[END_REF][START_REF] Pastur | A simple approach to global regime of random matrix theory[END_REF][START_REF] Capitaine | The largest eigenvalues of finite rank deformation of large Wigner matrices : convergence and nonuniversality of the fluctuations[END_REF] yields :

Var{û(t)} ≤ Kt 2 .
Since the entries of Y are assumed to have a fourth finite moment (which is the optimal condition for a CLT to hold for the same normalization), one can only use the generalized integration by part formula (cf. Proposition 3.2.2) up to the fourth cumulant, and this is not sufficient. In order to circumvent this issue, Lytova and Pastur perform a two-stage interpolation.

Since

V n = Var(u(t)) = Eů(t)ů(-t) = E(u(t) -û(t))(ů(-t) -ů(-t)) -Eû(t) ů(-t) = Eα(t)α(-t) -Eû(t) ů(-t), (3.2.14)
in order to perform the first interpolation, we shall introduce three terms T 1 -B, T 2 and ε 2 that arise from an application of the generalized integration by parts formula and write :

E(u(t) -û(t))α(-t) = it 1 0 ((T 1 -B) + T 2 + ε 2 )ds.
A deeper study on ε 2 will yield the second interpolation :

ε 2 = ε ′ 2 + it 1 0 ((T ′ 1 -B ′ ) + T ′ 2 + ε ′′ 2 )ds 1 .
Controlling successively the terms T 1 -B and T 2 , then the terms ε ′ 2 , T ′ 1 -B ′ , T ′ 2 and ε ′′ 2 will yield the master inequality

V n ≤ C 3 (t)V 1/2 n + C 8 (t),
where C ℓ (t) is a polynomial in |t| with nonnegative coefficients and degree ℓ ≥ 0. This inequality yields the conclusion.

First interpolation

Useful estimates

Denote C ℓ (t) a polynomial in |t| with nonnegative coefficients and degree ℓ. The coefficients of C ℓ (t) may change from line to line, however the degree is ℓ. As introduced in the precedent paragraph,

V n = Var(u(t)) = E[(u(t) -û(t))α(-t)] -Eû(t) ů(-t).
Notice that Var{u(t)} = Var{u(-t)}. We first give an estimate of Var[û(t)].

Proposition 3.2.3

The following estimates hold true

Var(û(±t)) ≤ C 2 (t) (3.2.15) E 1/2 |α(±t)| 2 ≤ C 0 V 1/2 n + C 1 (t). ( 3

.2.16)

Proof : By Nash-Poincaré inequality (see for instance [START_REF] Pastur | Eigenvalue distribution of large random matrices[END_REF]Chapter 7]) and 3.2.9, we have

Var[û(t)] ≤ i,j E ∂ û(t) ∂ Xij 2 ≤ C 2 (t) n E[Tr Û(t)R Û * (t) Ŷ ŶT ] ≤ C 2 (t) n E[Tr Ŷ ŶT ] ≤ C 2 (t).
This proves (3.2.15). (3.2.16) can be shown by Cauchy-Schwarz.

E 1/2 |α(t)| 2 ≤ √ 2E 1/2 (|ů n (t)| 2 + | ůn (t)| 2 ) ≤ √ 2 V n + C 2 (t) ≤ C 0 V 1/2 n + C 1 (t).
The following lemma is the key to estimate [Y T U (t)R

|[Y T U (t)Y ] kk (t)| ≤ α (Y αk ) 2 , (3.2.17) 0 ≤ ℓ ≤ 3, |D ℓ jk [Y T U (t)R 1/2 ] kj | ≤ C ℓ (t) n ℓ/2 α Y 2 αk (ℓ+1)/2 , (3.2.18) a, b ≥ 0, |D a pq D b jk [Y T U (t)R 1/2 ] kj | ≤ C a+b (t) n (a+b)/2 α Y 2 αk (b+1)/2 α Y 2 αq a/2 , (3.2.19) 1 ≤ ℓ ≤ 3, |D ℓ jk u(t)| ≤ C ℓ (t) n ℓ α Y 2 αk ℓ/2 . (3.2.20)
The proof is postponed to Appendix 3.5.3.

The following proposition gives an estimate of the right hand side of the terms in the precedent lemma.

Proposition 3.2.4 For ℓ = 1, 2, 3, 4, E| α Y 2 αk | ℓ ≤ C 0 .
The proof is in Appendix 3.5.5.

Introduction of the terms T 1 -B, T 2 and ε 2

We now treat the term E(u(t)û(t))α(-t). By considering the interpolating matrix

Y(s) = s 1/2 R 1/2 X √ n + (1 -s) 1/2 R 1/2 X √ n + A,
we have

u(t) -û(t) = 1 0 d ds u(t, s)ds = it √ n 1 0 Tr exp(itY s Y s T ) R 1/2 X 2 √ s - R 1/2 X 2 √ 1 -s Y T s ds + it √ n 1 0 Tr exp(itY s Y T s )Y s R 1/2 X T 2 √ s - R 1/2 XT 2 √ 1 -s ds = it √ n 1 0 Tr R 1/2 X √ s - R 1/2 X √ 1 -s Y T s U(t, s) ds. (3.2.21)
Hence the relation,

E[(u(t) -û(t))α(-t)] △ = it 1 0 [A n -B n ]ds, (3.2

.22)

where

A n = 1 √ ns n j,k=1 E{X jk Φ n }, B n = 1 n(1 -s) n j,k=1 E{ Xjk Φ n } with Φ n = [Y s T U (t, s)R 1/2 ] kj α(-t).
By the generalized integration by part formula (cf. Proposition 3.2.2) with p = 2, and beware that κ 1 = 0, (cf. Section 3.1.4) we have

A n = T 1 + T 2 + ε 2
where

T 1 = κ 2 √ sn n j,k E ∂ ∂X jk [Y T s U (t, s)R 1/2 ] kj α(-t) + κ 2 √ sn n j,k E [Y T s U (t, s)R 1/2 ] kj ∂ ∂X jk α(-t) , T 2 = κ 3 2 √ sn n j,k E ∂ 2 ∂(X jk ) 2 [Y T s U (t, s)R 1/2 ] kj α(-t) + κ 3 √ sn n j,k E ∂ ∂(X jk ) [Y T s U (t, s)R 1/2 ] kj ∂ ∂(X jk ) α(-t) + κ 3 2 √ sn n j,k E [Y T s U (t, s)R 1/2 ] kj ∂ 2 ∂(X jk ) 2 α(-t) and |ε 2 | ≤ C 0 µ 4 √ ns n j,k=1 sup |W |≤ηn √ n ∂ 3 ∂(X jk ) 3 E {Φ n (X)|X jk = W } .
The crux of this interpolation procedure is the partial cancelations that occur between the terms T 1 and B. The aim of the this section is to prove that

|Eα(t)α(-t)| = |t| 1 0 (A n -B n )ds ≤ C 3 (t)V 1/2 n + C 4 (t) + |t| 1 0 |ε 2 |ds. (3.2.23) Estimation for T 1 -B n Recall T 1 = κ 2 √ sn n j,k E ∂ ∂X jk [Y T s U (t, s)R 1/2 ] kj α(-t) + κ 2 √ sn n j,k E [Y T s U (t, s)R 1/2 ] kj ∂ ∂X jk α(-t) , B n = κ 2 n(1 -s) n j,k=1
E{ Xjk Φ n }.

In this section, we will show that

|T 1 -B n | ≤ C 2 (t) √ s + C 2 (t) √ 1 -s . ( 3

.2.24)

By the generalized integration by part,

B n = κ 2 (1 -s)n n j,k E ∂ ∂ Xjk [Y T s U (t, s)R 1/2 ] kj α(-t) + κ 2 (1 -s)n n j,k E [Y T s U (t, s)R 1/2 ] kj ∂ ∂ Xjk α(-t) ,
Notice that it is possible to use D jk (s) in the first term of the r.h.s. (see Section 3.2.1) but not in the second where we use instead D jk (see Remark 3.2.1). With this remark, we have

T 1 = κ 2 √ n n j,k E{D jk (s)[Y T s U (t, s)R 1/2 ] kj α(-t)} + κ 2 √ sn n j,k E{[Y T s U (t, s)R 1/2 ] kj D jk α(-t)}, B n = κ 2 √ n n j,k E{D jk (s)[Y T s U (t, s)R 1/2 ] kj α(-t)} + κ 2 (1 -s)n n j,k E{[Y T s U (t, s)R 1/2 ] kj Djk α(-t)}.
and using (3.2.9),

T 1 -B n = κ 2 √ sn n j,k E{[Y T s U (t, s)R 1/2 ] kj 2i(-t)[Y T U (-t)R 1/2 ] kj } - κ 2 (1 -s)n n j,k E{[Y T s U (t, s)R 1/2 ] kj 2i(-t)[ Ŷ T Û (-t)R 1/2 ] kj } △ = T 1a + T 1b .
We deal with T 1a :

|T 1a | ≤ C 1 (t) √ sn E[Tr(R 1/2 U(-t)YY T s U(t, s)R 1/2 )] ≤ C 1 (t) √ sn E 1/2 [Tr(RU(t, s) * Y s Y T s U(t, s))]E 1/2 [Tr(RU(-t) * YY T U(-t))] ≤ C 1 (t) √ s ,
and |T 1b | can be treated in a similar way. This ends the estimation of (3.2.24).

Estimation for T 2

Recall the expression of T 2 :

T 2 = κ 3 2 √ sn n j,k E ∂ 2 ∂(X jk ) 2 [Y T s U (t, s)R 1/2 ] kj α(-t) + κ 3 √ sn n j,k E ∂ ∂(X jk ) [Y T s U (t, s)R 1/2 ] kj ∂ ∂(X jk ) α(-t) + κ 3 2 √ sn n j,k E [Y T s U (t, s)R 1/2 ] kj ∂ 2 ∂(X jk ) 2 α(-t) △ = T 2a + T 2b + T 2c .
In this section, we will show that

|T 2 | ≤ C 2 (t)V 1/2 n + C 3 (t) √ s . ( 3

.2.25)

We start with T 2a . Using the differentiation formula (3.2.11) and remark 3.2.1, we obtain

T 2a = √ sκ 3 2n 1/2 n j,k E D 2 jk (s)[Y T s U (t, s)R 1/2 ] kj α(-t) = 3iκ 3 √ s n 3/2 j,k E ([R 1/2 U (t, s)R 1/2 ] jj * [Y T s U (t, s)R 1/2 ] kj )α(-t) - 3κ 3 √ s n 3/2 j,k E [Y T s U (t, s)Y s ] kk * [Y T s U (t, s)R 1/2 ] kj * [R 1/2 U (t, s)R 1/2 ] jj α(-t) - κ 3 √ s n 3/2 j,k E [Y T s U (t, s)R 1/2 ] kj * [Y T s U (t, s)R 1/2 ] kj * [Y T s U (t, s)R 1/2 ] kj α(-t) .
In order to handle T 2a , we rely on the following proposition whose proof is in Appendix 3.5.4.

Proposition 3.2.5 Let Z be a random variable with (E|Z| 2 ) 1/2 ≤ K Z , then the following estimates hold true, for

t 1 , t 2 , t 3 ∈ R E{ j,k [R 1/2 U (t 1 , s)R 1/2 ] jj [Y T s U (t 2 , s)R 1/2 ] kj Z} ≤ C 0 n 3/2 K Z , (3.2.26) j,k E{[Y T s U (t 1 , s)Y s ] kk [Y T s U (t 2 , s)R 1/2 ] kj [R 1/2 U (t 3 , s)R 1/2 ] jj Z} ≤ C 0 n 3/2 K Z , (3.2.27) j,k E{[Y T s U (t 1 , s)R 1/2 ] kj [Y T s U (t 2 , s)R 1/2 ] kj [Y T s U (t 3 , s)R 1/2 ] kj } ≤ C 0 n 3/2 K Z . (3.2.28) Remark 3.2.3
The same result holds true for the terms

[R 1/2 U (t 1 )R 1/2 ] jj , [Y T s U (t 2 , s)R 1/2 ] kj , i.e. E{ j,k [R 1/2 U (t 1 )R 1/2 ] jj [Y T U (t 2 )R 1/2 ] kj Z} ≤ C 0 n 3/2 K Z .
We obtain similar results for the other two terms. In the sequel, we shall freely use the proposition with U (t) and U (t, s).

Replacing Z by α(-t) with

K Z = C 0 V 1/2 n + C 1 (t) (cf. (3.2.16
)), and taking into account the fact that every convolution product adds a factor t, we end up with :

|T 2a | ≤ C 1 (t)V 1/2 n + C 2 (t) + C 2 (t)V 1/2 n + C 3 (t) + C 2 (t)V 1/2 n + C 3 (t) ≤ C 2 (t)V 1/2 n + C 3 (t).
(3.2.29)

Recall that

T 2b = κ 3 2 √ sn j,k E{D jk [Y T s U (t, s)R 1/2 ] kj D jk α(-t)} = κ 3 2n 1/2 j,k E{D jk (s)[Y T s U (t, s)R 1/2 ] kj D jk u(-t)} = itκ 3 n 3/2 j,k E{[R 1/2 U (t, s)R 1/2 ] jj × [Y T U (t)R 1/2 ] kj } - tκ 3 n 3/2 j,k E{[Y T U (t)R 1/2 ] kj × [Y T s U (t, s)Y s ] kk * [R 1/2 U (t, s)R 1/2 ] jj } - tκ 3 n 3/2 j,k E{[Y T U (t)R 1/2 ] kj × [Y T s U (t, s)R 1/2 ] kj * [Y T s U (t, s)R 1/2 ] kj }.
We use again Proposition 3.2.5 with Z = 1 and obtain

|T 2b | ≤ C 2 (t). (3.2.30)
Taking advantage from (3.2.12), we handle the term T 2c as follows.

T 2c = κ 3 2 √ ns j,k E{[Y T s U (t, s)R 1/2 ] kj D 2 jk α(-t)} = itκ 3 n 3/2 √ s j,k E{[Y T s U (t, s)R 1/2 ] kj × [R 1/2 U (-t)R 1/2 ] jj } - tκ 3 n 3/2 √ s j,k E{[Y T s U (t, s)R 1/2 ] kj × [Y T U (-t)Y ] kk * [R 1/2 U (-t)R 1/2 ] jj } - tκ 3 n 3/2 √ s j,k E{[Y T s U (t, s)R 1/2 ] kj × [Y T U (-t)R 1/2 ] kj * [Y T U (-t)R 1/2 ] kj }.
By Proposition 3.2.5 with Z = 1, as there is one convolution product, a factor t appears :

|T 2c | ≤ C 2 (t) √ s . ( 3
|V n | ≤ C 3 (t)V 1/2 n + C 4 (t) + |t| 1 0 |ε 2 |ds. (3.2.32)
It now remains to handle ε 2 .

Decomposition of ε 2

Now we will evaluate ε 2 . Recall that

|ε 2 | ≤ C 0 √ ns j,k sup |W |≤ηn √ n ∂ 3 ∂W 3 E {Φ(W )|X jk = W } with Φ(W ) = E{[Y T s U (t 1 , s)R 1/2 ] kj α(-t)|X jk = W }.
In particular, we will show that

|ε 2 (s)| ≤ C 7 (t) √ s (3.2.33)
As Y jk is bounded (cf. Section 3.1.4) and Ŷ has all moments, we can change the order between integration and the derivation and expand the third derivative :

D 3 jk [Y T s U (t 1 , s)R 1/2 ] kj α(-t) =D 3 jk [Y T s U (t, s)R 1/2 ] kj α(-t) + 3 ℓ=1 C ℓ 3 D 3-ℓ jk [Y T s U (t, s)R 1/2 ] kj D ℓ jk α(-t) .
Applying (3.2.13), we have

|ε 2 | ≤ C 0 √ n j,k S jk + |ε ′ 2 |, (3.2.34)
where

S jk = sup |W |≤ηn √ n E{D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj α(-t)|X jk = W } , (3.2.35) |ε ′ 2 | = C 0 √ sn j,k sup |W |≤ηn √ n E 3 ℓ=1 C ℓ 3 D 3-ℓ jk [Y T s U (t, s)R 1/2 ] kj D ℓ jk α(-t)|X jk = W . ( 3
C ℓ 3 D 3-ℓ jk [Y T s U (t, s)R 1/2 ] kj D ℓ jk u(-t) = 3 ℓ=1 C ℓ 3 s (3-ℓ)/2 D 3-ℓ jk (s)[Y T s U (t, s)R 1/2 ] jk D ℓ jk u(-t) ≤ C 3 (t) n 3/2 3 ℓ=1 E   α [Y s ] 2 αk (4-ℓ)/2 α [Y αk ] 2 ℓ/2   .
(3.2.37)

Now we will show that for all ℓ = 1, 2, 3,

E   α [Y s ] 2 αk (4-ℓ)/2 α (Y αk ) 2 ℓ/2   ≤ C 0 . (3.2.38) When ℓ = 2, E α [Y s ] 2 αk α (Y αk ) 2 ≤ 2E α [Y αk ] 2 + [ Ŷαk ] 2 α (Y αk ) 2 ≤ C 0 ,
where the last inequality follows from the hypothesis that

E|x ij | 4 < ∞. When ℓ = 1, E    α [Y s ] 2 αk 3/2 α (Y αk ) 2 1/2    ≤ E 1/2    α [Y s ] 2 αk 2    E 1/2 α [Y s ] 2 αk α (Y αk ) 2 ≤ C 0 .
And ℓ = 3 can be treated similarly. (

1 √ n E 3 ℓ=1 C ℓ 3 D 3-ℓ jk [Y T s U (t, s)R 1/2 ] kj D ℓ jk α(-t)|X jk = W ≤ C 3 (t) n 2 (1 + |Y jk | 4 ) ≤ C 3 (t) n 2 . ( 3

Second interpolation

Recall that

S jk = sup |W |≤ηn √ n E{D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj α(-t)|X jk = W } .
In order to estimate S jk , we perform an interpolation procedure on the quantity α(-t). With the same method as (3.2.22),

E D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj α(-t)|X jk = W =E D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj 1 0 ds 1 d ds 1 u(-t, s 1 )|X jk = W = - it √ n 1 0 E D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj Tr R 1/2 X √ s 1 - R 1/2 X √ 1 -s 1 Y T s 1 U(-t, s 1 ) ds 1 |X jk = W △ = -it 1 0 (A ′ 0 -B ′ )ds 1 ,
where

A ′ 0 = 1 √ n E D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj Tr X √ s 1 Y T s 1 U(-t, s 1 )R 1/2 |X jk = W , B ′ = 1 √ n E D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj Tr X √ 1 -s 1 Y T s 1 U(-t, s 1 )R 1/2 |X jk = W .
By distinguishing (p, q) = (j, k), we have

A ′ 0 = 1 √ n E D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj Tr X √ s 1 Y T s 1 U(-t, s 1 )R 1/2 |X jk = W = 1 √ ns 1 ′ p,q E X pq [Y T s 1 U (-t, s 1 )R 1/2 ] qp D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X jk = W + X jk √ ns 1 E [Y T s 1 U (-t, s 1 )R 1/2 ] jk D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X jk = W △ = A ′ + A ′ 0a ,
where ′ pq stands for the sum over the indices {p, q} = {j, k}. We first prove that

|A ′ 0a | ≤ C 3 (t)W √ s 1 n 3/2 .
By Lemma 3.2.2,

|A ′ 0a | ≤ C 3 (t) W √ s 1 n 2 E    α [Y s 1 ] 2 αk 1/2 α [Y s ] 2 αk 2 |X jk = W    ≤ C 3 (t)W √ s 1 n 2 E    α (Y αk ) 2 + Ŷ 2 αk 5/2 |X jk = W    ≤ C 3 (t)W √ s 1 n 2 E    α (Y αk ) 2 5/2 |X jk = W    + C 3 (t) W √ s 1 n 2 E    α ( Ŷαk ) 2 5/2 |X jk = W    ≤ C 3 (t)W √ s 1 n 2 E 1/2    α (Y αk ) 2 3 |X jk = W    E 1/2    α (Y αk ) 2 2 |X jk = W    + C 3 (t)W n 2 √ s 1 (a) ≤ C 3 (t)W √ s 1 n 2
where (a) is from Proposition 3.2.4. Hence

|A ′ 0a | ≤ C 3 (t)W √ s 1 n 2 ≤ C 3 (t)η n √ s 1 n 3/2 .
Plugging this and (3.2.40) into (3.2.34), we have

|ε 2 | ≤ C 4 (t) + C 0 |t| √ n sup |W |≤ηn √ n 1 0 A ′ -B ′ ds 1
where

A ′ = 1 √ ns 1 ′ p,q E X pq [Y T s 1 U (-t, s 1 )R 1/2 ] qp D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X jk = W B ′ = 1 √ ns 1 ′ p,q E Xpq [Y T s 1 U (-t, s 1 )R 1/2 ] qp D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X jk = W .
By integration by part formula, (cf. Proposition 3.2.2), we have

A ′ = T ′ 1 + T ′ 2 + ǫ ′′ 2 (3.2.41)
where

T ′ 1 = κ 2 √ ns 1 ′ p,q ED pq [Y T s 1 U (-t, s 1 )R 1/2 ] qp D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X jk = W T ′ 2 = κ 3 √ s 1 n ′ p,q ED 2 pq [Y T s 1 U (-t, s 1 )R 1/2 ] qp D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X jk = W B ′ = κ 2 n(1 -s 1 ) ′ p,q E Dpq [Y T s 1 U (-t, s 1 )R 1/2 ] qp D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X jk = W , and
|ε ′′ 2 | ≤ C 0 √ ns 1 ′ p,q sup |Z|≤ηn √ n ∂ ∂Z 3 E [Y T s 1 U t,s 1 R 1/2 ] qp D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X pq = Z, X jk = W .
All concentrations should be paid to estimate these terms. As in the first interpolation, B ′ cancels with T ′ 1 . However, as more terms appear, we need a more delicate method to estimate these terms. One must understand all terms in the development.

Estimation of T ′ 1 -B ′
We now handle the term T ′ 1 -B ′ . From (3.2.41) where some cancelation occurs. Recall the expression of T ′ 1 and B ′ :

T ′ 1 = κ 2 √ ns 1 ′ p,q ED pq [Y T s 1 U (-t, s 1 )R 1/2 ] qp D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X jk = W B ′ = κ 2 n(1 -s 1 ) ′ p,q E Dpq [Y T s 1 U (-t, s 1 )R 1/2 ] qp D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X jk = W .
We shall prove that

|W |≤ηn √ n |t| √ n 1 0 |T ′ 1 -B ′ |ds 1 ≤ C 5 (t). (3.2.42)
As in the first interpolation along with Remark 3.2.1,

D pq [Y T s 1 U (-t, s 1 )R 1/2 ] qp = √ s 1 D pq (s 1 )[Y T s 1 U (-t, s 1 )R 1/2 ] qp Dpq [Y T s 1 U (-t, s 1 )R 1/2 ] qp = √ 1 -s 1 D pq (s 1 )[Y T s 1 U (-t, s 1 )R 1/2 ] qp .
Then we have

1 √ n (T ′ 1 -B ′ ) = κ 2 √ s 1 n ′ p,q E [Y T s 1 U (-t, s 1 )R 1/2 ] qp D pq D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X jk = W - κ 2 √ 1 -s 1 n ′ p,q E [Y T s 1 U (-t, s 1 )R 1/2 ] qp Dpq D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X jk = W △ = T ′ 1a + T ′ 1b .
The goal in the following is to show that

|T ′ 1a | ≤ C 4 (t) √ s 1 (3.2.43) |T ′ 1b | ≤ C 4 (t) √ 1 -s 1 .
This implies immediately (3.2.42). T ′ 1a and T ′ 1b can be handled similarly. We shall focus on T ′ 1a . Adding the term corresponding to (p, q) = (j, k) does not affect the order of magnitude of T ′ 1a since by (3.2.19) and Proposition 3.2.4,

κ 2 √ s 1 n j,k E [Y T s 1 U (-t, s 1 )R 1/2 ] kj D jk D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X jk = W ≤ C 4 (t) √ s 1 n 3 j,k E    α [Y s 1 ] 2 αk 1/2 α [Y s ] 2 αk 5/2 |X jk = W    ≤ C 4 (t) √ s 1 n 1 + W 6 n 3 .
Now we will disclose all terms in the development of

[Y T s 1 U (-t, s 1 )R 1/2 ] qp D pq (s)D 3 jk [Y T s U (t, s)R 1/2 ] kj .
Since the factor before the sum is of order n -1 , a rough estimate of each term of the sum as done before will not be sufficient here. One needs to look carefully to the terms depending on p and q in the derivative

D pq D 3 jk (s)[Y T s U (t, s)R 1/2
] kj and to understand how they combine with [Y T s 1 U (t, s 1 )R 1/2 ] pq when summing up. We first need to understand all terms in

D 3 jk (s)[Y T s U (t, s)R 1/2 ] jk .
Based upon the terms appearing in the second derivative

D 2 jk (s)[Y T s U (t, s)R 1/2 ] kj (cf. (3.2.11
)) and the first order derivatives, all terms in

D 3 jk (s)[Y T s U (t, s)R 1/2
] kj are :

L 1 = [R 1/2 U (t, s)R 1/2 ] jj * [R 1/2 U (t, s)R 1/2 ] jj L 2 = [R 1/2 U (t, s)R 1/2 ] jj * [Y T s U (t, s)R 1/2 ] kj * [Y T s U (t, s)R 1/2 ] kj L 3 = [R 1/2 U (t, s)R 1/2 ] jj * [R 1/2 U (t, s)R 1/2 ] jj * [Y T s U (t, s)Y s ] kk L 4 = [R 1/2 U (t, s)R 1/2 ] jj * [Y T s U (t, s)R 1/2 ] kj * [Y T s U (t, s)R 1/2 ] kj * [Y T s U (t, s)Y s ] kk L 5 = [R 1/2 U (t, s)R 1/2 ] jj * [R 1/2 U (t, s)R 1/2 ] jj * [Y T s U (t, s)Y s ] kk * [Y T s U (t, s)Y s ] jj L 6 = [Y T s U (t, s)R 1/2 ] kj * [Y T s U (t, s)R 1/2 ] kj * [Y T s U (t, s)R 1/2 ] kj * [Y T s U (t, s)R 1/2 ] kj .
In order to understand the action of D pq upon all these terms, it is sufficient to understand the action of D pq (s) over the three recurring terms :

[R 1/2 U (t, s)R 1/2 ] jj , [Y T s U (t, s)R 1/2 ] kj , [Y T s U (t, s)Y s ] kk .
Computation yields :

D pq [R 1/2 U (t, s)R 1/2 ] jj = 2i √ s[Y s U (t, s)R 1/2 ] qj * [R 1/2 U (t, s)R 1/2 ] jp , D pq [Y T s U (t, s)R 1/2 ] kj = √ sδ q,k [R 1/2 U (t, s)R 1/2 ] pj + i √ s[Y T s U (t, s)Y s ] qk * [R 1/2 U (t, s)R 1/2 ] pj + √ s[Y T s U (t, s)R 1/2 ] qj * [Y T s U (t, s)R 1/2 ] kp , D pq [Y T s U (t, s)Y s ] kk = 2 √ sδ q,k [R 1/2 U (t, s)R 1/2 ] kp + 2i √ s[Y T s U (t, s)R 1/2 ] kp * [Y T s U (t, s)Y s ] qk . Now consider the sets B C [Y T s U (t, s)R 1/2 ] qj * [R 1/2 U (t, s)R 1/2 ] jp [R 1/2 U (t, s)R 1/2 ] jj δ q,k [R 1/2 U (t, s)R 1/2 ] pj [Y T s U (t, s)R 1/2 ] kj [Y T s U (t, s)Y s ] qk * [R 1/2 U (t, s)R 1/2 ] pj [Y T s U (t, s)Y s ] kk [Y T s U (t, s)R 1/2 ] qj * [Y T s U (t, s)R 1/2 ] kp δ q,k [R 1/2 U (t, s)R 1/2 ] kp [Y T s U (t, s)R 1/2 ] kp * [Y T s U (t, s)Y s ] qk
and introduce the set

G = {B(p, q) * C 1 (j, k), B(p, q) * C 1 (j, k) * C 2 (j, k), B(p, q) * C 1 (j, k) * C 2 (j, k) * C 3 (j, k)} , where B(p, q) ∈ B, C i ∈ C for i = 1, 2, 3.
Up to a constant, the generic terms G that appear in T ′ 1a are

G = p,q [Y T s 1 U (t, s 1 )R 1/2 ] pq × g,
where g ∈ G and

T ′ 1a = κ 2 n √ s 1 g∈G E{G|X jk = W }.
We now assert that :

∀g ∈ G, sup |W |≤ηn √ n E{|G||X jk = W } ≤ nC 4 (t).
The factor C 4 (t) comes from the fact that the maximum power of t in the estimate above is give by the maximum number of convolution products which is 4.

Computations are similar and we only handle one term as an example. Taking

G = p,q [Y T s 1 U (t, s 1 )R 1/2 ] qp × [Y T s U (t, s)R 1/2 ] kp * [Y T s U (t, s)Y s ] qk * [R 1/2 U (t, s)R 1/2 ] jj * [R 1/2 U (t, s)R 1/2 ] jj * [Y T s U (t, s)Y s ] kk , (3.2.44) we have p,q [Y T s 1 U (t, s 1 )R 1/2 ] qp × [Y T s U (t 1 , s)R 1/2 ] kp [Y T s U (t 2 , s)Y s ] qk ×[R 1/2 U (t 3 , s)R 1/2 ] jj [R 1/2 U (t 4 , s)R 1/2 ] jj [Y T s U (t 5 , s)Y s ] kk = [Y T s U (t 1 , s)RU (t, s)Y s 1 Y T s U (t 2 , s)Y s ] kk × [R 1/2 U (t 3 , s)R 1/2 ] jj [R 1/2 U (t 4 , s)R 1/2 ] jj [Y T s U (t 5 , s)Y s ] kk ≤ C 0 α [Y T s U (t 1 , s)RU (t, s)Y s 1 ] kα [Y T s U (t 2 , s)Y s ] αk ×   β [Y s,βk ] 2   ≤ C 0 α   β [Y s 1 ] 2 βα   1/2   β [Y s ] 2 βα   1/2   β [Y s ] 2 βk   2 . (3.2.45)
This kind of term can be handled as before and its expectation is of order n. Hence the estimate

sup |W |≤ηn √ n |t| 1 0 |T ′ 1a |ds 1 ≤ C 5 (t). (3.2.46)
This ends the proof of (3.2.42).

Estimation of T ′ 2

In this section, we shall prove that

sup |W |≤ηn √ n |t| √ n 1 0 |T ′ 2 |ds 1 ≤ C 6 (t). (3.2.47)
From (3.2.41), we have

1 √ n T ′ 2 = κ 3 √ s 1 n ′ p,q ED 2 pq [Y T s 1 U (-t, s 1 )R 1/2 ] qp D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X jk = W = κ 3 √ s 1 n ′ p,q E D 2 pq (s 1 )[Y T s 1 U (-t, s 1 )R 1/2 ] qp D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X jk = W + κ 3 n ′ p,q E D pq (s 1 )[Y T s 1 U (-t, s 1 )R 1/2 ] qp D pq D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X jk = W + κ 3 √ s 1 n ′ p,q E [Y T s 1 U (-t, s 1 )R 1/2 ] qp D 2 pq D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X jk = W △ = T ′ 2a + T ′ 2b + T ′ 2c .
Beware that we use the relation D pq = √ s 1 D pq (s 1 ) and apply the differentiation opera-

tor D pq (s 1 ) to the quantity [Y T s 1 U (-t, s 1 )R 1/2 ] qp but not the quantity [Y T s U (t, s)R 1/2 ] kj which depends on Y s but not on Y s 1 .
Notice that by Lemma 3.2.2,

sup |W |≤ηn √ n E D 3 jk [Y T s U (t, s)R 1/2 ] kj 2 |X jk = W 1/2 ≤ C 3 (t) n 3/2 .
Hence, T ′ 2a can be handled in the same way as the term T 2a with the replacement of αn by

D 3 jk [Y T s U (t, s)R 1/2 ] kj . This yields |T ′ 2a | ≤ C 5 (t).
Now we will handle the term T ′ 2b which is by far the most delicate.

T ′ 2b = κ 3 n ′ p,q E D pq (s 1 )[Y T s 1 U (-t, s 1 )R 1/2 ] qp D pq D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X jk = W
In order to proceed, we shall perform the same analysis as for the term in T ′ 1a . The analysis is based on a precise description of the generic term that appears when expanding the derivative

D pq (s 1 )[Y T s 1 U (t, s 1 )R 1/2 ] pq ], the derivative D pq D 3 jk [Y T s U (t, s)R 1/2
] jk and their product. Up to a constant, the generic terms are

G ′ = p,q A × g, where A ∈ A = [R 1/2 U (-t, s 1 )R 1/2 ] pp , [Y T s 1 U (-t, s 1 )Y s 1 ] qq * [U (-t, s 1 )] pp , [Y T s 1 U (-t, s 1 )R 1/2 ] pq * [Y T s 1 U (-t, s 1 )R 1/2 ] pq
and g ∈ G with G defined in (3.2.49). Then we can show without difficulties that,

|T ′ 2b | = κ 3 √ s 1 n 3/2 A∈A,g∈G E{G ′ |X jk = W } ≤ C 5 (t).
(3.2.48)

We make an example as usual :

G ′ = p,q [Y T s 1 U (t, s 1 )R 1/2 ] qp * [Y T s 1 U (t, s 1 )R 1/2 ] qp × [Y T s U (t, s)R 1/2 ] kp * [Y T s U (t, s)Y s ] qk * [R 1/2 U (t, s)R 1/2 ] jj * [R 1/2 U (t, s)R 1/2 ] jj * [Y T s U (t, s)Y s ] kk , (3.2.49) we have E p,q [Y T s 1 U (t, s 1 )R 1/2 ] qp [Y T s 1 U (t 1 , s 1 )R 1/2 ] qp × [Y T s U (t 2 , s)R 1/2 ] kp [Y T s U (t 3 , s)Y s ] qk ×[R 1/2 U (t 4 , s)R 1/2 ] jj [R 1/2 U (t 5 , s)R 1/2 ] jj [Y T s U (t 6 , s)Y s ] kk ≤ C 0 E p,q α [Y s 1 ] 2 αq 1/2 [Y T s 1 U (t 1 , s 1 )R 1/2 ] pq [Y T s U (t 2 , s)R 1/2 ] kp [Y T s U (t 3 , s)Y s ] qk [Y T s U (t 6 , s)Y s ] kk ≤ √ nC 0 E [Y T s U (t 2 , s)RU (t 1 , s 1 )Y s 1 Y T s U (t 3 , s)Y s ] kk α [Y s ] 2 αk ≤ C 0 n 3/2 ,
where the last inequality follows from the same argument as T ′ 1a . The degree 5 of the polynomial comes from the fact that we have 5 convolutions. Thus we get (3.2.48). Now we will evaluate T ′ 2c .

|T ′ 2c | = κ 3 √ s 1 n ′ p,q E [Y T s 1 U (-t, s 1 )R 1/2 ] qp D 2 pq D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X jk = W ≤ κ 3 √ s 1 n ′ p,q E [Y T s 1 U (-t, s 1 )R 1/2 ] qp 2 |X jk = W 1/2 × ′ p,q E D 2 pq D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj 2 |X jk = W 1/2 ≤ C 5 (t) √ s 1 ,
where the last inequality follows from Lemma 3.2.2 and Proposition 3.2.4. This ends the proof of (3.2.47).

Estimation of ǫ ′′

Recall that Changing the order between the derivation and the integration yields :

|ǫ ′′ 2 | ≤ C 0 √ ns 1 ′ p,q sup |Z|≤ηn √ n ∂ ∂Z 3 E [Y T s 1 U t,s 1 R 1/2 ] qp D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X pq = Z, X jk = W . ( 3
1 n ∂ ∂Z 3 E [Y T s 1 U (t, s 1 )R 1/2 ] qp D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X pq = Z, X jk = W = 1 n E ∂ ∂Z 3 [Y T s 1 U (t, s 1 )R 1/2 ] qp D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X pq = Z, X jk = W = 1 n E 3 ℓ=0 C ℓ 3 ∂ ℓ ∂Z ℓ [Y T s 1 U (t, s 1 )R 1/2 ] qp ∂ ℓ ∂Z 3-ℓ D 3 jk (s)[Y T s U (t, s)R 1/2 ] kj |X pq = Z, X jk = W (a) ≤ C 0 n 4 E   3 ℓ=0 C ℓ (t) α [Y s 1 ] 2 αq (ℓ+1)/2 C 6-ℓ (t) α [Y s ] 2 αk (7-ℓ)/2 |X pq = Z, X jk = W   (b) ≤ C 6 (t) n 2 3 ℓ=0 1 + |Z| √ n ℓ 1 + |W | √ n 7-ℓ
.

where

V n ≤ C 3 (t)V 1/2 n + C 8 (t).

This implies that

V n ≤ C 0 (1 + |t| 4 ) 2 .

More interpolations

For future use, we prove the following proposition which will serve in the next section.

Eα(t) △ = it 1 0 (A ′′ n -B ′′ n )ds (3.2.52)
where

A ′′ n = 1 √ ns j,k E{X jk [Y T s U R 1/2 ] kj }, B ′′ n = 1 n(1 -s) j,k E{ Xjk [Y T s U R 1/2 ] kj }.

By Proposition 3.2.2, we have

A ′′ n = T ′′ 1 + T ′′ 2 + ε ′′′ 2 (3.2.53)
where

T ′′ 1 = κ 2 √ sn n j,k E ∂ ∂X jk [Y T s U (t, s)R 1/2 ] kj , T ′′ 2 = κ 3 2 √ sn n j,k E ∂ 2 ∂(X jk ) 2 [Y T s U (t, s)R 1/2 ] kj , B ′′ n = κ 2 √ sn n j,k E ∂ ∂ Xjk [Y T s U (t, s)R 1/2 ] kj , |ε ′′′ 2 | ≤ C 0 µ 4 √ ns n j,k=1 sup |W |≤ηn √ n ∂ 3 ∂(X jk ) 3 E [Y T s U (t, s)R 1/2 ] kj |X jk = W . Then A ′′ n -B ′′ n = T ′′ 1 -B ′′ n + T ′′ 2 + ε ′′′ 2 .
As [Y T s U (t, s)R 1/2 ] kj is a function of X s , we have

1 √ s D jk [Y T s U (t, s)R 1/2 ] kj = D jk (s)[Y T s U (t, s)R 1/2 ] kj = 1 √ 1 -s Djk [Y T s U (t, s)R 1/2 ] kj , which implies T ′′ 1 -B ′′ n = 0. (3.2.54)
For T ′′ 2 , by (3.2.11),

T ′′ 2 = √ sκ 3 2n 1/2 n j,k E D 2 jk (s)[Y T s U (t, s)R 1/2 ] kj = 3iκ 3 √ s n 3/2 j,k E [R 1/2 U (t, s)R 1/2 ] jj * [Y T s U (t, s)R 1/2 ] kj - 3κ 3 √ s n 3/2 j,k E [Y T s U (t, s)Y s ] kk * [Y T s U (t, s)R 1/2 ] kj * [R 1/2 U (t, s)R 1/2 ] jj - κ 3 √ s n 3/2 j,k E [Y T s U (t, s)R 1/2 ] kj * [Y T s U (t, s)R 1/2 ] kj * [Y T s U (t, s)R 1/2 ] kj .
By Proposition 3.2.5 with Z = 1 and taking two convolutions into accnout, we get Another useful interpolation is between the real Gaussian matrices and the complex Gaussian matrices which will serve in the calculation of the bias. Proposition 3.2.7 Denote Xr (resp. Xc ), the standard real (resp. complex) Gaussian random matrix. Consider Proof : As before, consider the interpolation matrix

|T ′′ 2 | ≤ C 2 (t). (3.2.55) For ε ′′′ 2 , by (3.2.18), |ε ′′′ 2 | ≤ C 3 (t) n 2 j,k sup |W |≤ηn √ n E    α [Y αk ] 2 2 |X jk = W    ≤ C 3 (t). ( 3
Ŷr = 1 √ n R 1/2 n Xr + A n Ŷc = 1 √ n R 1/
X s = √ s Xr + √ 1 -s Xc , Y s = 1 √ n R 1/2 n X s + A n ,
and relative term U(t, s). Write

Eû r (t) -Eû c (t) = 1 0 d ds ETre itYsY * s ds = it 1 0 ETr e itYsY * s 1 2 √ s Ŷr - 1 2 √ 1 -s Ŷc Y * s ds + it 1 0 ETr e itYsY * s Y s 1 2 √ s Ŷr - 1 2 √ 1 -s Ŷc * ds.
The two terms are similar and we shall focus on the first term.

We have

1 0 ETr e itYsY * s 1 √ s Ŷr - 1 √ 1 -s Ŷc Y * s ds △ = 1 0 ( Â(s) -B(s))ds where Â(s) △ = 1 √ ns j,k E{ Xr,jk [Y * s U (t, s)R 1/2 ] kj }, B(s) △ = 1 n(1 -s) j,k E{ Xc,jk [Y * s U (t, s)R 1/2 ] kj }.
For a complex Gaussian variable X and a function Φ(X) of class C 1 , by integration by parts, we have

E[XΦ(X)] = E|X| 2 E ∂Φ(X) ∂X , (3.2.57) E[XΦ(X)] = E|X| 2 E ∂Φ(X) ∂X . (3.2.58)
With the precedent formula, we have

Â(t, s) △ = 1 √ ns j,k E ∂ ∂ Xr,jk [Y * s U (t, s)R 1/2 ] kj , B(t, s) △ = 1 n(1 -s) j,k E ∂ ∂ Xc,jk [Y * s U (t, s)R 1/2 ] kj . From (3.2.8), ∂ ∂ Xr,jk [Y * s U (t, s)R 1/2 ] kj = √ s √ n [R 1/2 U (t, s)R 1/2 ] jj + i √ s √ n [Y * s U (t, s)Y s ] kk * [R 1/2 U (t, s)R 1/2 ] jj + i √ s √ n [Y * s U (t, s)R 1/2 ] kj * [Y * s U (t, s)R 1/2 ] kj .
The same method yields that

∂ ∂ Xc,jk [Y * s U (t, s)R 1/2 ] kj = √ 1 -s √ n [R 1/2 U (t, s)R 1/2 ] jj + i √ 1 -s √ n [Y * s U (t, s)Y s ] kk * [R 1/2 U (t, s)R
Then Â(t, s) -B(t, s) = i n j,k [Y * s U (t, s)R 1/2 ] kj * [Y * s U (t, s)R 1/2 ] kj .
With the same argument as Lemma 3.2.2 and take the convolution into account, (see also the argument for

T 1a in (3.2.24)) | Â(t, s) -B(t, s)| ≤ C 1 (|t|),
and we have

|Eû r (t) -Eû c (t)| ≤ C 2 (|t|).

Proof of Theorem 3.1.1 : CLT for non-analytic functionals

We return back to the model

Y n = 1 √ n R 1/2 n X n ,
and shall establish the CLT for the linear statistics N (f ) defined in (3.1.3). Recently, the study of N (f ) when f is analytic is investigated in Najim [START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF]. Recall the definition of K n (z 1 , z 2 ) in (3.1.6) and define

Θ 0,n △ = t′ n (z 1 ) t′ n (z 2 ) ( tn (z 1 ) -tn (z 2 )) 2 - 1 (z 1 -z 2 ) 2 (3.3.1) Θ 1,n △ = ∂ 2 ∂z 1 ∂z 2 K n (z 1 , z 2 ). (3.3.2)
Recall the following Central Limit Theorem.

∀t i ∈ R, 1 ≤ i ≤ d, d LP ( • u n (t 1 ), • • • , • u n (t d )), (ψ n (t 1 ), • • • , ψ n (t d )) ---→ n→∞ 0, (3.3.4)
where ψ n denotes the sequence of centered Gaussian random variables with the covariance

Cov(ψ n (t i ), ψ n (t j )) = - 1 4π 2 e it i z 1 e it j z 2 Θ n (z 1 , z 2 )dz 1 dz 2 , 1 ≤ i, j ≤ d.
Thus the finite dimensional convergence of f (t)ů n (t) is achieved.

Condition (ii). Now we will prove the tightness of ů(x) over a certain interval [-T, T ]. From [START_REF] Billingsley | Convergence of probability measures[END_REF]Theorem 12.3], it suffices to verify there exists t 0 ∈ [-T, T ] such that :

E|ů n (t 0 )| 2 < ∞, and sup n,t 1 ,t 2 ∈[-T,T ] E ůn (t 1 ) -ůn (t 2 ) t 1 -t 2 2 < ∞.
By Proposition 3.2.3 and 3.2.6,

E|ů n (t 0 )| 2 ≤ C 4 (|t 0 |) + Var[û n (t 0 )] < ∞. Since ůn (t 1 ) -ůn (t 2 ) t 1 -t 2 = 1 2iπ Cn e izt 1 -e izt 2 t 1 -t 2 M 1 n (z)dz,
we have

E ůn (t 1 ) -ůn (t 2 ) t 1 -t 2 2 = 1 4π 2 C 1 C 2 e iz 1 t 1 -e iz 1 t 2 t 1 -t 2 e iz 2 t 1 -e iz 2 t 2 t 1 -t 2 E M 1 n (z 1 )M 1 n (z 2 ) dz 1 dz 2 ,
where C 1 and C 2 are two contours non-overlapping both of which contain the interval (3.3.3). By a simple calculus, we have

sup t 1 ,t 2 ∈[-T,T ] e it 1 z 1 -e it 2 z 1 t 1 -t 2 = sup t 1 ,t 2 ∈[-T,T ] z 1 1 0 e it 2 z 1 (1-s) e it 1 z 1 s ds < ∞.
Moreover, from Proposition 3.3.1,

sup z 1 ∈C 1 ,z 2 ∈C 2 EM 1 n (z 1 )M 1 n (z 2 ) ≤ sup z 1 ∈C 1 ,z 2 ∈C 2 VarM 1 n (z 1 ) VarM 1 n (z 2 ) < ∞. Therefore, sup n,t 1 ,t 2 ∈[-T,T ] E ůn (t 1 ) -ůn (t 2 ) t 1 -t 2 2 < ∞.
The tightness of the process ů(x) is achieved which implies Condition (ii).

Condition (iii). It remains to estimate the variance. By the inverse formula of Fourier transform,

N i=1 f (λ i ) = R f (x)u n (x)dx.
By Theorem 3.1.3, we have

Var 1/2 [ f (t)u n (t)] ≤ C 0 | f (t)|(1 + |t| 4 ) (3.3.5)
which is integrable on R by hypothesis. Now we will use the argument of meta-model to show that

Var 1/2 [ f (t)ψ n (t)] ≤ C 0 | f (t)|(1 + |t| 4 )
where ψ n ia a Gaussian variable (cf. (3.3.4)).

Introduction of the meta-model

The initial idea of meta-model is from [START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF]. With the extra parameter M ∈ N, consider the

N M × nM matrix R n (M ) =     R n • • • 0 . . . . . . . . . 0 • • • R n    
where R n (M ) is the block matrix whose diagonal is composed of M matrices R n and 0 elsewhere. Consider now the random matrix

Y n (M ) = 1 √ nM R 1/2 n (M )X n (M ),
where X n (M ) is an M N × M n matrix with i.i.d. random entries with the same distribution as the X ij 's and satisfying (3.1.1).

Denote M 1 n,M (z) = N M j=1 1 λ j -z -N M j=1 E 1 λ j -z where (λ i ) 1≤i≤N M are eigenvalues of Y n (M )Y n * (M ).
Cov[M 1 n,M (z 1 ), M 1 n,M (z 2 )] ----→ M →∞ Θ n (z 1 , z 2 )
where

Θ(z 1 , z2) = 2Θ 0,n (z 1 , z 2 ) + κ 4 Θ 1,n (z 1 , z 2 ).
With these notations at hand, consider the set of matrices :

{R n (M ) : M ≥ 1, R n is an N × n matrix which satisfies Assumption 3.1.3}.

According to [18, Chapter 2, Section 7], a sequence of Gaussian process is tight if and only if it is relatively compact in distribution. Since R n (M ) = R n for all M ≥ 1, we have

sup M ≥1,N,n→∞ R n (M ) = sup N,n→∞
Var 1/2 [ f (t)u M n (t)] ≤ C 0 | f (t)|(1 + |t| 4 ). Since | f (t)|(1 + |t| 4 ) is supposed integrable in R, in particular f (t)u M n (t)
is tight, hence relatively compact in distribution. As the distribution L(ψ n ) of the Gaussian process ψ n is the limit in M of the distribution of u M n , belongs to the closure of {u M n : M ≥ 1} n,N →∞ , which is relatively compact. Finally, L(ψ n ) is included in a compact set, hence is relatively compact. In particular, the family of gaussian processes (ψ n ) is tight and consider

h K (x) = |x| 2 ∧ K, Eh K (ψ n (t)) = lim M →∞ Eh K (ů M n (t)) ≤ lim sup M →∞ E|ů M n (t)| 2 ≤ C 2 0 (1 + |t| 4 ) 2 .
Let K → ∞ and by monotone convergence theorem, we get

Var 1/2 [ f (t)ψ n (t)] ≤ C 0 | f (t)|(1 + |t| 4 ).
All conditions in Lemma 3.1.2 being verified, we conclude that

d LP (N 1 n (f ), Z 1 n (f )) -----→ N,n→∞ 0
where the variance of Z 1 n (f ) will be determined in the next section.

Computation of the covariance

In this section, we will interpret the covariance with the real integral. As f is no more analytic, the existence of the contour integral will not be ensured in this case. By Proposition 3.3.1, and the inverse formula of Fourier transform, the covariance for two functions

f 1 , f 2 satisfying (3.1.8) is Cov   N j=1 f 1 (λ j ), N j=1 f 2 (λ j )   = Cov   R N j=1 e iλ j x 1 f1 (x 1 )dx 1 , R N j=1 e iλ j x 2 f2 (x 2 )dx 2   = - 1 4π 2 R R f1 (x 1 ) f2 (x 2 )dx 1 dx 2 C 1 C 2 e iz 1 x 1 e iz 2 x 2 Θ n (z 1 , z 2 )dz 1 dz 2 + o(1).
Then by Proposition 3.3.1, recall the definitions (3.3.1), (3.3.2), we have

C 1 C 2 e iz 1 x 1 e iz 2 x 2 Θ n (z 1 , z 2 )dz 1 dz 2 = - 1 2π 2 C 1 C 2 e iz 1 x 1 e iz 2 x 2 Θ 0,n (z 1 , z 2 )dz 1 dz 2 - κ 4 4π 2 C 1 C 2 e iz 1 x 1 e iz 2 x 2 Θ 1,n (z 1 , z 2 )dz 1 dz 2 . (3.3.6)
The first double integral is in [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF]Eq. (1.20)] and we recall some elements of the proof. By integration by parts twice, it suffices to study

C 1 C 2 e iz 1 x 1 e iz 2 x 2 t′ (z 1 ) t′ (z 2 ) ( t(z 1 ) -t(z 2 )) 2 dz 1 dz 2 = - C 1 C 2 x 1 x 2 e iz 1 x 1 e iz 2 x 2 log( t(z 1 ) -t(z 2 ))dz 1 dz 2 .
Fixing z 2 ∈ C 2 , we deform C 1 and get :

C 1 e iz 1 x 1 e iz 2 x 2 log( t(z 1 ) -t(z 2 ))dz 1 = Sn e ix 1 s 1 e iz 2 x 2 log( t(s 1 ) -t(z 2 )) -log( t(s 1 ) -t(z 2 )) ds 1 ,
where S n is the support of the measure Fn associated with tn . With the same method, we develop the double integral according to z 2 , we get

C 1 C 2 e iz 1 x 1 e iz 2 x 2 log( t(z 1 ) -t(z 2 ))dz 1 dz 2 = Sn Sn e ix 1 s 1 e is 2 x 2 log | t(s 1 ) -t(s 2 )| 2 -log | t(s 1 ) -t(s 2 )| 2 ds 1 ds 2 = 2
Sn Sn

e ix 1 s 1 e ix 2 s 2 [log | t(s 1 ) -t(s 2 )| -log | t(s 1 ) -t(s 2 )|]ds 1 ds 2 . (3.3.7) Then - 1 π 2 R R f1 (x 1 ) f2 (x 2 )dx 1 dx 2 Sn Sn ix 1 ix 2 e ix 1 s 1 e ix 2 s 2 log t(s 1 ) -t(s 2 ) t(s 1 ) -t(s 2 ) ds 1 ds 2 = - 1 π 2 Sn Sn log t(s 1 ) -t(s 2 ) t(s 1 ) -t(s 2 ) ds 1 ds 2 R R f1 (x 1 ) f2 (x 2 )ix 1 ix 2 e ix 1 s 1 e ix 2 s 2 dx 1 dx 2 = - 1 π 2 Sn Sn f ′ 1 (s 1 )f ′ 2 (s 2 ) log t(s 1 ) -t(s 2 ) t(s 1 ) -t(s 2 ) ds 1 ds 2 = 1 2π 2 Sn Sn f ′ 1 (s 1 )f ′ 2 (s 2 ) log 1 + 4Im[ t(s 1 )]Im[ t(s 2 )] | t(s 1 ) -t(s 2 )| 2 ds 1 ds 2 . (3.3.8)
The same method applies to deal with the second integral in (3.3.6) :

C 1 C 2 e iz 1 x 1 e iz 2 x 2 Θ 1,n (z 1 , z 2 )dz 1 dz 2 = -x 1 x 2 C 1 C 2 e iz 1 x 1 e iz 2 x 2 K n (z 1 , z 2 )dz 1 dz 2 .
where K n (z 1 , z 2 ) is defined in (3.1.6). It is easy to check that tn (s) = tn (s) and T n (s) = T n (s).

Fixing z 2 and we get

C 1 e iz 1 x 1 e iz 2 x 2 K n (z 1 , z 2 )dz 1 = Sn e is 1 x 1 e iz 2 x 2 (K n (s 1 , z 2 ) -K † n (s 1 , z 2 ))ds 1 ,
where

K † n (s 1 , z 2 ) is defined in (3.1.7
). The development for z 2 yields that

C 1 C 2 e iz 1 x 1 e iz 2 x 2 K n (z 1 , z 2 )dz 1 dz 2 = 2
Sn Sn

e is 1 x 1 e is 2 x 2 Re(K n (s 1 , s 2 ) -K † n (s 1 , s 2 ))ds 1 ds 2 ,
which leads to

- κ 4 4π 2 R R f1 (x 1 ) f2 (x 2 )dx 1 dx 2 C 1 C 2 e iz 1 x 1 e iz 2 x 2 Θ 1,n (z 1 , z 2 )dz 1 dz 2 = - κ 4 2π 2 Sn Sn f ′ 1 (s 1 )f ′ 2 (s 2 )Re(K n (s 1 , s 2 ) -K † n (s 1 , s 2 ))ds 1 ds 2 .
These terminate the computation of the expression (3.1.9).

Proof of Theorem 3.1.2 : Estimation of the bias

In this paragraph, we will calculate the bias. This issue has been studied in several papers as [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF][START_REF] Vallet | Improved subspace estimation for multivariate observations of high dimension : The deterministic signals case[END_REF][START_REF] Haagerup | A new application of random matrices : Ext(C * red (F 2 )) is not a group[END_REF]. We first recall the bias calculated in [START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF] for analytic functions. Proposition 3.4.1 Under the same settings as Theorem 3.1.2, consider the deterministic vector

N 2 n (f ) = (N 2 n (f 1 ), • • • , N 2 n (f k ))
where N 2 n (f ) is defined in (3.1.5) and the vector

Z 2 n (f ) △ = (Z 2 n (f 1 ), • • • , Z 2 n (f k ))
where

EZ 2 n (f j ) = 1 2iπ f j (z)B n (z)dz, for 1 ≤ j ≤ k, with B n (z) = B 1,n + κ 4 B 2,n ,
|Z 2 n (f )| < ∞ and N 2 n (f ) -Z 2 n (f ) -----→ N,n→∞ 0.
Denote the bias

N 2 n (f ) = N E f (x)F YY T (dx) -N R f (x)F n (dx),
where F n is the measure associated with deterministic equivalent t n (cf. (3.1.1)). Then consider

N 2 n (f ) △ = N 2 n ′ (f ) + N 2 n ′′ (f ) + N 2 n ′′′ (f )
where

N 2 n ′ (f ) = N E f (x)F YY T (dx) -N E f (x)F Ŷr ŶT r (dx) (3.4.1)
N 2 n ′′ (f ) = N E f (x)F Ŷr ŶT r (dx) -N E f (x)F Ŷc Ŷ * c (dx) (3.4.2) N 2 n ′′′ (f ) = N E f (x)F Ŷc Ŷ * c (dx) -N f (x)F n (dx), (3.4.3)
and Ŷr (resp. Ŷc ) is the matrix associated with standard real (resp. complex) Gaussian entries.

Outline of the proof :

1. For complex Gaussian bias N 2 n ′′′ (f ), following [START_REF] Vallet | Improved subspace estimation for multivariate observations of high dimension : The deterministic signals case[END_REF], we first establish the inequality

|N (E mc (z) -t(z))| ≤ 1 n C 12 (|z|)C 17 (|Imz| -1 ),
where C ℓ (t) is the polynomial of degree ℓ on |t| with positive coefficients and mc is the Stieljes transform associated with complex Gaussian entries (cf. Proposition 3.4.2). Then from the idea in [START_REF] Haagerup | A new application of random matrices : Ext(C * red (F 2 )) is not a group[END_REF], for ϕ ∈ C 18 c (R, R) where C 18 c (R, R) denotes the set of function of class C 18 with compact support, we show that

|N 2 n ′′′ (ϕ)| ≤ K n |(1 + D) 18 ϕ(x)|(1 + |x|) 12 dx,
where the operator D = d dx . Finally we approximate f by a function

f (x) ∈ C 18 c (R, R) to get N 2 n ′′′ (f ) ---→ n→∞ 0 (cf. Theorem 3.4.1).
2. Recall u n (t) = Tre itYY T and ûc (t) = Tre it Ŷc Ŷ * c . Using Fourier transform, the expression of the bias Eu n (t) -Eû c (t) is deduced from Proposition 3.4.1 and 3.4.2. Then Proposition 3.2.6 and 3.2.7 will provide the domination

|Eu n (t) -Eû c (t)| ≤ |Eu n (t) -Eû r (t)| + |Eû r (t) -Eû c (t)| ≤ C 4 (|t|), in order to obtain |N 2 n ′ (f ) + N 2 n ′′ (f ) -B| -----→ N,n→∞ 0,
where B is the bias to be calculated (cf. (3.4.44)).

Complex Gaussian bias

We start the proof with the estimation

N 2 n ′′′ (f ). Denote mc (z) = 1 N Tr( Ŷc Ŷ * c -zI N ) -1 and mc (z) = 1 N Tr( Ŷ * c Ŷc -zI n ) -1 .
In [START_REF] Vallet | Improved subspace estimation for multivariate observations of high dimension : The deterministic signals case[END_REF][START_REF] Dumont | On the Capacity Achieving Covariance Matrix for Rician MIMO Channels : An Asymptotic Approach[END_REF][START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF], it is known that under complex Gaussian case, the bias

N E mc (z) = N t(z) + 1 n ξ N (z)
where ξ N is analytic in C\R + and satisfies

|ξ N (z)| ≤ (|z| + C) k P (|Imz| -1 ), for each z ∈ C +
where C is a constant, k ∈ N and P is a polynomial with positive coefficients independent of N . We shall specify the degree of polynomial P and the integer k which are useful in the estimation of the bias. Following the idea in [START_REF] Vallet | Improved subspace estimation for multivariate observations of high dimension : The deterministic signals case[END_REF], we have Proposition 3.4.2 Denote the bias in Gaussian case by

M 2 c (z) = N E mc (z) -N t(z). Then | M 2 c (z)| ≤ 1 n C 12 (|z|)C 17 (|Imz| -1 ),
where

C ℓ (|z|) (resp. C ℓ (|Imz| -1 )) is a polynomial in |z| (resp. |Imz| -1
) with positive coefficient of degree ℓ.

Proof : For reason of readability, in the proof, we write Y instead of Ŷc to denote the standard complex Gaussian vector and denote the relative terms in complex Gaussian case by

Q = ( Ŷc Ŷ * c -zI) -1 and mc = 1 n Tr( Ŷ * c Ŷc -zI) -1 .
We have the derivatives

∂[Q] pq ∂X ij = - 1 √ n [QR 1/2 ] pi [Y * Q] jq , ∂[Q] pq ∂X ij = - 1 √ n [R 1/2 Q] iq [QY * ] pj . (3.4.4)
and define also intermediate matrices

S = -[z(I + E mc (z)R)] -1 , (3.4.5)
S = τ I, (3.4.6)
and τ = -

1 z(1 + 1 n ETrRQ)
.

From QYY * = I + zQ, we have

δ p,q + zEQ pq = E i,j [Q] pi Y ij Y * jq = 1 √ n E i,j [QR 1/2 ] pi X ij Y qj . (3.4.7)
By (3.2.58) with Φ(X) = [QR 1/2 ] pi Y qj and (3.4.4), we have

E[QR 1/2 ] pi X ij Y qj = E ∂[QR 1/2 ] pi ∂X ij Y qj + E ∂Y qj ∂X ij [QR 1/2 ] pi = - 1 √ n E[RQ] ii [QY ] pj Y qj + δ i,q √ n E[QR] pq .
Then (3.4.7) can be rewritten as :

E[QY Y * ] pq = - 1 n E{[QY Y * ] pq TrRQ} + E[QR] pq . (3.4.8) Let ω = 1 n TrRQ.
Noticing that ω = ω + Eω, and resolving (3.4.8) according to E[QY Y * ] pq , we have

δ p,q + zEQ pq = 1 1 + Eω E[QR] pq - 1 1 + Eω E{[QY Y * ] pq ω}.
Therefore we get

I + ∆(z) = EQ(-zI -zτ (z)R) (3.4.9)
where

∆(z) = 1 1 + Eω E [Q(z)YY * (ω -Eω)] . (3.4.10)
Lemma 3.4.1 It holds that

z(E mc (z) -τ (z)) = - 1 n Tr∆(z).
Proof : Taking trace in (3.4.9) and using the fact that mc (z) = c n mc -1-cn z , we have

- 1 n Tr∆(z) = z n ETrQ + z n τ (z)ETrQR + c n = zE mc (z) -zτ (z).
Then the right hand side in (3.4.9) is

EQ(-zI -zτ (z)R) = EQS -1 + z(E mc (z) -τ (z))EQ(z)R.
Using Lemma 3.4.1 and Eq. (3.4.9), we have

EQ(z) -S(z) = ∆(z)S(z) + 1 n (Tr∆(z))EQ(z)RS(z), which implies that N E mc (z) -TrS(z) = Tr∆(z)S(z) + 1 n Tr∆(z)Tr[EQ(z)RS(z)]. (3.4.11)
Now we will show that Proposition 3.4.3 For z ∈ C + , we have

|ETrQ(z) -TrS(z)| ≤ 1 n C 3 (|z|)C 7 (|Imz| -1 ), (3.4.12
)

|ETrRQ(z) -TrRS(z)| ≤ 1 n C 3 (|z|)C 7 (|Imz| -1 ), (3.4.13
)

|n( mc (z) -τ (z))| ≤ 1 n C 2 (|z|)C 5 (|Imz| -1 ) (3.4.14)
Proof : We first apply the following preliminary result which is based on Nash-Poincaré inequality (cf. [74, Proposition 2.1.6]).

Lemma 3.4.2 Consider an N × N matrix A satisfying A < ∞. For all z ∈ C + , we have

Var [TrQ(z)A] ≤ A 2 C 1 (|z|)C 4 (|Imz| -1 ).
The proof is in Appendix 3.5.6. We now complete the proof of Proposition 3.4.3. We shall estimate the two terms in (3.4.11). Remark that

Im[Eω] = 1 2in [ETrRQ(z) -ETrRQ(z)] > 0, we have 1 |z(1 + Eω)| ≤ 1 |Imz| .
This along with the inequality Q(z) ≤ 1 |Imz| lead to

1 n Tr∆(z)Tr[EQ(z)RS(z)] ≤ K |Imz| 2 |Tr∆(z)| . (3.4.15)
From the expression of ∆ in (3.4.10) and by Lemma 3.4.2 with A = I and A = R, we have

|Tr∆| ≤ z (1 + Eω) E [(TrQ -ETrQ)(ω -Eω)] ≤ |z| 2 |Imz| |E [(TrQ -ETrQ)(ω -Eω)]| ≤ 1 n C 3 (|z|)C 5 (|Imz| -1 )
.

Plugging the precedent estimate into (3.4.15), we have

1 n Tr∆(z)Tr[EQ(z)RS(z)] ≤ 1 n C 3 (|z|)C 7 (|Imz| -1 ). (3.4.16)
With the same approach, since S(z) ≤ |Imz| -1 , we have

|Tr[∆(z)S(z)]| ≤ 1 n C 3 (|z|)C 6 (|Imz| -1 ). (3.4.17)
Gathering (3.4.11), (3.4.16) and (3.4.17), (3.4.12) is proved. The same method applies for (3.4.13). By Lemma 3.4.1 and the expression of ∆ in (3.4.10),

n| mc (z) -τ (z)| = 1 z(1 + Eω) E [Q(z)YY * (ω -Eω)] ≤ |z| |Imz| |E [(TrQ -ETrQ)(ω -Eω)]| ≤ 1 n C 2 (|z|)C 5 (|Imz| -1 )
Thus we achieve Proposition 3.4.3.

Denote

ε n (z) = n(E mc -τ ) εn (z) = ETrRQ -TrRS.
We remark that

n(E mc (z) -t(z)) = n(τ -t) + ε n (z),
ETrRQ -TrRT = TrRS -TrRT + εn (z).

Using the expression of T (resp. τ ) in (3.1.2) (resp. (3.4.5)) and the fact that for two matrix A and B, A -B = -A(A -1 -B -1 )B, we obtain that

n(E mc (z) -t(z)) ETrRQ -TrRT = D 0 (z) n(E mc (z) -t(z)) ETrRQ -TrRT + ε n (z) εn (z) , (3.4.18)
where

D 0 (z) = 0 zv 0 (z) zṽ 0 (z) 0
with v 0 (z) and ṽ0 (z) defined by

v 0 (z) = τ (z) t(z) ṽ0 (z) = 1 n Tr[RSRT] = 1 n Tr[RSTR].
Then we have

(I -D 0 (z)) n(E mc (z) -t(z)) ETrRQ -TrRT = ε n (z) εn (z) . (3.4.19)
Now we will resolve this system. We first evaluate the determinant of the system. We introduce the matrices

D(z) = 0 v(z) |z| 2 ṽ(z) 0 (3.4.20) D ′ (z) = 0 v ′ (z) |z| 2 ṽ′ (z) 0 (3.4.21)
with v(z), ṽ(z) v ′ (z) and ṽ′ (z) defined by

v(z) = t(z) t(z) , ṽ(z) = 1 n Tr[RT(z)T(z) * R], (3.4.22)
v ′ (z) = τ (z)τ (z) , ṽ′ (z) = 1 n Tr[RS(z)S(z) * R]. (3.4.23)
We have the following proposition which will be proved in Appendix 3.5.7.

Proposition 3.4.4

There exists a strictly positive constant η such that

det(I -D(z)) ≥ K|Imz| 4 (η 2 + |z| 2 ) 2 , (3.4.24)
for each z ∈ C + . Moreover, there exists an integer N 0 and polynomials C 12 (|z|) and C 16 (|Imz| -1 ) such that for each N > N 0 ,

det(I -D ′ (z)) ≥ K|Imz| 4 (η 2 + |z| 2 ) 2 , (3.4.25)
for each element of z in the set E defined by

E = {z ∈ C + , 1 - 1 n 2 C 12 (|z|)C 16 (|Imz| -1 ) > 0}.
Finally, for each N > N 0 and z ∈ E,

|det(I -D 0 (z))| ≥ det(I -D(z)) det(I -D ′ (z)) ≥ K|Imz| 4 (η 2 + |z| 2 ) 2 . (3.4.26)
We complete the estimation of the bias. From (3.4.19),

n(E mc (z) -t(z)) = 1 det(I -D 0 ) [ε(z) + zv 0 ε(z)].
For z ∈ E, by Proposition 3.4.3 and

|v 0 | ≤ 1 |Imz| 2 , |E mc (z) -t(z)| ≤ 1 n 2 C 8 (|z|)C 13 (|Imz| -1 ). (3.4.27) For z ∈ C + \E, as 1 -1 n 2 C 12 (|z|)C 16 (|Imz| -1 ) > 0, 2 < 2 n 2 C 12 (|z|)C 16 (|Imz| -1 ). Then |n(E mc (z) -t(z))| ≤ 2n |Imz| ≤ 1 n C 12 (|z|)C 17 (|Imz| -1 ). (3.4.28)
Combining (3.4.27) and (3.4.28), we have

|n(E mc (z) -t(z))| ≤ 1 n C 12 (|z|)C 17 (|Imz| -1 ).
Next step consists in applying Proposition 3.4.2 to estimate N 2 n ′′′ (f ).

Theorem 3.4.1 Recall the complex Gaussian bias

N 2 ′′′ (f ) = N E R f (x)F Ŷc Ŷ * c (dx) - R f (x)F n (dx) .
For a function f of class C 18 , we have

|N 2 ′′′ (f )| ---→ n→∞ 0.
Proof : The proof is split into two parts :

1. For a function ϕ ∈ C 18 c (R, R) where C 18 c (R, R) is the set of compactly supported functions of class C 18 . 2. Approximate f by a function f ∈ C 18 c (R, R) Denote Mc (z) = (E mc (z) -t n (z)
) and recall the deterministic equivalent t n with the associated measure F n . By the inverse Stieljes transform,

F Ŷc Ŷ * c (dx) = lim y→0 + - 1 π Im[ mc (x + iy)]dx .
In particular, we have :

For ϕ ∈ C 18 c (R, R), E R ϕ(x)F Ŷc Ŷ * c (dx) = lim y→0 + - 1 π E R Im[ϕ(x) mc (x + iy)]dx , (3.4.29)
R ϕ(x)F n (dx) = lim y→0 + - 1 π R Im[ϕ(x)t n (x + iy)]dx . ( 3
E R ϕ(x)F Ŷc Ŷ * c (dx) - R ϕ(x)F n (dx) ≤ 1 π lim sup y→0 + R [ϕ(x) Mc (x + iy)]dx . (3.4.31)
For z ∈ C + and p ∈ N, define

I p (z) = 1 (p -1)! ∞ 0 Mc (z + t)t p-1 e -t dt. (3.4.32)
Note that I p (z) is well defined because Mc (z) is uniformly bounded when z ∈ C + . Also, it is easy to check that I p (z) is an analytic function of z and the first derivative is given by

I ′ p (z) = 1 (p -1)! ∞ 0 M ′ c (z + t)t p-1 e -t dt.
Then by partial integration we get

I ′ 1 (z) = [ Mc (z + t)e -t ] ∞ 0 + ∞ 0 Mc (z + t)e -t dt = -Mc (z) + I 1 (z).
In the same way for p ≥ 2,

I ′ p (z) = 1 (p -1)! ∞ 0 M ′ c (z + t)t p-1 e -t dt = - 1 (p -1)! ∞ 0 Mc (z + t)((p -1)t p-2 -t p-1 )e -t dt = -I p-1 (z) + I p (z). (3.4.33)
We have then :

R ϕ(x) Mc (x + iy)dx = R ϕ(x)I 1 (x + iy) - R ϕ(x)I ′ 1 (x + iy)dx = R ϕ(x)I 1 (x + iy) - R ϕ ′ (x)I 1 (x + iy)dx = R ((1 + D)ϕ)(x)I 1 (x + iy)dx,
where D = d dx . Using (3.4.33), we iterate the precedent procedure and we obtain : for all p ≥ 1,

R ϕ(x) Mc (x + iy) = R ((1 + D) p ϕ)(x)I p (x + iy)dx.
Hence by (3.4.31), we have for all p ∈ N ;

E R ϕ(x)F Ŷc Ŷ * c (dx) - R ϕ(x)F n (dx) ≤ 1 π lim sup y→0 + R ((1 + D) p ϕ)(x)I p (x + iy)dx .
(3.4.34) Next, we will use Proposition 3.4.2 to show that for p = 18, and z ∈ C + , one has

|I 18 (z)| ≤ C 12 (|z|) n 2 (3.4.35)
where C 12 (|z|) is a polynomial on |z| of degree 12.

We fix w ∈ C + . To prove (3.4.35), we apply Cauchy integral theorem to the function

F (z) = 1 17! Mc (w + z)z 17 e -z ,
which is analytic in the half-plane Imz > -Imw. Hence for r > 0,

F (z)dz ≤ 2 17!|Imw| r 0 |r + it| 17 e -r dt ≤ 2 17!|Imw| (2r) 17 re -r ---→ r→0 0.
Therefore,

I 18 (z) = 1 17! ∞ 0 Mc (z + t)t 17 e -t dt = lim r→∞ [0,r] F (t)dt = lim r→∞ [0,r+ir] F (t)dt = 1 17! ∞ 0 Mc (z + (1 + i)t)[(1 + i)t] 17 e -(1+i)t (1 + i)dt.
By Proposition 3.4.3,

| Mc (z)| ≤ 1 n 2 C 12 (|z|)C 17 (|Imz| -1 ). Then |I 18 (z)| ≤ 1 n 2 ∞ 0 C 12 (|z + (1 + i)t|)C 17 (|Imz + t| -1 )( √ 2t) 17 e -t √ 2dt ≤ 1 n 2 ∞ 0 C 12 (|z|)e -t √ 2dt ≤ C 12 (|z|) n 2 .
This proves (3.4.35). Combining (3.4.34) and (3.4.35), we have

E R ϕ(x)F Ŷc Ŷ * c (dx) - R ϕ(x)F n (dx) ≤ 1 π lim sup y→0 + R ((1 + D) 18 ϕ)(x)I 18 (x + iy)dx ≤ K n 2 R |(1 + D) 18 ϕ(x)|(1 + |x|) 12 dx.
In particular, for any function ϕ

(x) ∈ C 18 c (R, R), N E R ϕ(x)F Ŷc Ŷ * c (dx) -N R ϕ(x)F n (dx) -----→ n,N →∞ 0. (3.4.36)
Next task is to extend (3.4.36) for a function f of class C 18 . Let ε > 0. By [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF]Appendix],

there exists a compact K 1 such that for all k ∈ N, P(min

i d( λi , K 1 ) ≥ ε) = o(n -k ), λi eigenvalues of Ŷc Ŷ * c . (3.4.37)
From [START_REF] Bai | No Eigenvalues Outside the Support of the Limiting Spectral Distribution of Large Dimensional Sample Covariance Matrices[END_REF], the support of the limiting measure t n is a compact which is denoted by K 2 . Now we take a compact interval K such that

K 1 ∪ K 2 ⊂ K. Consider a function f ∈ C 18 c (R, R) defined by f (x) = f (x), when x ∈ K. As f (x) ∈ C 18 c (R, R), by (3.4.36), we have N R E f (x)F Ŷc Ŷ * c (dx) -N R f (x)F n (dx) -----→ N,n→∞ 0. The proof of Theorem 3.4.1 is completed if one can show N R Ef (x)F Ŷc Ŷ * c (dx) -N R E f (x)F Ŷc Ŷ * c (dx) -N R f (x)F n (dx) + N R f (x)F n (dx) -----→ N,n→∞ 0. (3.4.38)
In particular, we will show that

N E R f (x)F Ŷc Ŷ * c (dx) -N E R f (x)F Ŷc Ŷ * c (dx) -----→ N,n→∞ 0, (3.4.39)
N R f (x)F n (dx) -N R f (x)F n (dx) = 0. (3.4.40)
To prove (3.4.39), we have

N E R f (x)F Ŷc Ŷ * c (dx) -N E R f (x)F Ŷc Ŷ * c (dx) ≤ N E K f (x)F Ŷc Ŷ * c (dx) -N E K f (x)F Ŷc Ŷ * c (dx) + N E R\K f (x)F Ŷc Ŷ * c (dx) + N E R\K f (x)F Ŷc Ŷ * c (dx) ≤ N E R\K f (x)F Ŷc Ŷ * c (dx) + N E R\K f (x)F Ŷc Ŷ * c (dx) ,
and

N E R\K f (x)F Ŷc Ŷ * c (dx) = N E R f (x)I x / ∈K F Ŷc Ŷ * c (dx) ≤ N 2 f ∞ P( λmax / ∈ K) (a) ≤ Kn -1
where λmax is the largest eigenvalue of Ŷc Ŷ * c and (a) follows from (3.4.37) and the fact that f is bounded in R (cf. Remark 3.1.1). Same argument holds true for f and this proves (3.4.39).

From the definition of

K 2 , R f (x)F n (dx) - R f (x)F n (dx) = K f (x)F n (dx) - K f (x)F n (dx) = 0.
(3.4.40) is proved. This ends the proof of Theorem 3.4.1.

Bias for non-analytic functionals

Now we can terminate the computations of the bias. Recall that

N 2 n (f ) = N E f (x)F YY T (dx) -N R f (x)F n (dx) = N 2 n ′ (f ) + N 2 n ′′ (f ) + N 2 n ′′′ (f ),
where By Fourier transform,

N 2 n ′ (f ) (resp. N 2 n ′′ (f ), N 2 n ′′′ (f )) is defined in (3.
N 2 n ′ (f ) + N 2 n ′′ (f ) = E f (x)F YY T (dx) -E f (x)F Ŷc Ŷ * c (dx) = E f (x)u(x)dx -E f (x)û c (x)dx. From Proposition 3.4.1, for z ∈ C + , |N (Em n (z) -t n (z)) -B n | ---→ n→∞ 0, (3.4.42)
where B n is defined in Proposition 3.4.1. By Proposition 3.4.3, the complex Gaussian bias satisfies

|N (E mc (z) -t n (z))| ---→ n→∞ 0. ( 3
|N (Em n (z) -E mc (z)) -B n | ---→ n→∞ 0.
As z → e itz is analytic and lim sup n |B n | < ∞, by Propostion 3.4.1 we have then

Eu n (t) -Eû c (t) - 1 2iπ e izt B n (z)dz ---→ n→∞ 0.
Moreover, by Proposition 3.2.6 and 3.2.7,

| f (t)(Eu n (t) -Eû c (t))| ≤ | f (t)(Eu n (t) -Eû r (t))| + | f (t)(Eû r (t) -Eû c (t))| ≤ | f (t)|C 4 (|t|)
which is integrable. By the argument of meta-model (see Section 3.3.1), we can show that

f (t) e izt B n (z)dz ≤ | f (t)|C 4 (|t|),
which is integrable in R.

The dominated convergence theorem concludes that

N 2 n ′ (f ) + N 2 n ′′ (f ) - 1 2iπ f (t)e izt B n (z)dzdt ---→ n→∞ 0. (3.4.44) (3.4.41) and (3.4.44) yield that
N 2 n (f ) - 1 2iπ f (t)e izt B n (z)dzdt ---→ n→∞ 0.
The last task is to calculate C f (t)e izt B n (z)dzdt. From [11, Section 5] and Proposition 3.4.1, B n (z) is uniformly bounded over the contour C. With a similar method in Section 3.3.2,

1 2iπ e izt B n (z)dz = 1 π Sn e ist Im(B n (s))ds,
where B n (s) is defined in (3.1.10). We conclude that

1 2iπ f (t)e izt B n (z)dzdt = 1 π Sn f (t)Im(B n (t))dt.
This terminates the computations of the bias. We first recall the following proposition in [START_REF] Kallenberg | Foundations of mordern Probability[END_REF].

Appendices

Proposition 3.5.1 [50, Theorem 4.28] Let ξ, ξ n , η T and η T n be random elements in a metric space (S, ρ) such that η T n D -→ η T as n → ∞ for fixed T , and moreover η T D -→ ξ when T → ∞.

Then ξ n → ξ holds under the further condition

lim T →∞ lim sup n→∞ E[ρ(η T n , ξ n ) ∧ 1] = 0. (3.5.1)
Taking

η T n = T -T φn (t)dt, ξ n = R φn (t)dt.
As φn (t) → ψ(t) and φn (t) is tight for all t ∈ [-T, T ],

η T n D ---→ n→∞ η T △ = T -T ψ(t)dt. Since Var 1/2 [ψ(t)] ≤ K(t) where K(t) is integrable over R, T -T ψ(t)dt D ----→ T →∞ R ψ(t)dt.
Now we verify Condition (3.5.1) :

η T n -ξ n -E(η T n -ξ n ) = ∞ T φn (x)dx + -T -∞ φn (x)dx.
The two terms are similar and we treat the first term. By Cauchy-Schwarz,

E ∞ T φn (x)dx 2 = E ∞ T ∞ T φ1n (x)φ n (x 2 )dx 1 dx 2 ≤ ∞ T Var 1/2 (φ n (x))dx 2 . Since Var 1/2 (φ n (x)) ≤ K(t) which is integrable in R, we have lim T →∞ lim sup n→∞ E ∞ T φn (x)dx 2 = 0.
All conditions in Proposition 3.5.1 are satisfied and we conclude that

φ(x)dx D ---→ n→∞ ψ(t)dt.

Proof of Lemma 3.2.1

For an N ×n matrix Y, By applying Duhamel formula (cf. Proposition 3.2.1) with U (Y+H) = exp(it(Y + H)(Y + H) T ) and U (Y) = exp(itYY T ), we have :

U (Y + H) -U (Y) = i t 0 exp(i(t -s)YY T )(YH T + HY T + YY T ) exp(is(Y + H)(Y + H) T ))ds.
Taking H = hE jk where E jk is the elementary matrix, we get :

D jk [U (t)] mn = i([Y T U (t)] km * [U (t)] jn + [U (t)] mj * [Y T U (t)] kn ).
The rest can be calculated similarly.

Proof of Lemma 3.2.2

It is worth noticing that for a unitary matrix U, we have for

α 1 = α 2 , k U kα 1 Ūkα 2 = 0,
and k |U kα 1 | 2 = 1.
We first prove (3.2.17) :

|[Y T U (t)Y ] kk (t)| ≤ | α 1 Y α 1 k [U (t)Y ] α 1 k | ≤ α 1 (Y α 1 k ) 2 1/2 α 1 |[U (t)Y ] α 1 k | 2 1/2 ≤ α 1 (Y α 1 k ) 2 1/2 α 1 ,α 2 ,α 3 [U (t)] α 1 α 2 [ Ū (t)] α 1 α 3 Y α 2 k Y α 3 k 1/2 ≤ α 1 (Y α 1 k ) 2 1/2 α 1 ,α 2 =α 3 |U α 1 α 2 | 2 (Y α 2 k ) 2 1/2 ≤ α (Y αk ) 2 .
For (3.2.18), when ℓ = 0, it is a simple application of Cauchy-Schwarz. By using (3.2.7)-(3.2.10) and iterating the derivatives, along with the precedent estimate, we have (3.2.18).

(3.2.20) is a direct consequence of (3.2.9) and (3.2.18). Now we will prove (3.2.19). For q = k, by (3.2.7)-(3.2.10),

D pq D b jk [Y T U (t)R 1/2 ] kj = iD b jk ([R 1/2 U (t)R 1/2 ] jp * [Y T U (t)Y ] qk +[Y T U (t)R 1/2 ] qj * [Y T U (t)R 1/2 ] kp ).
Then noticing that all terms are composed of

[Y T U (t)R 1/2 ] a 1 b 1 , [Y T U (t)Y 1/2 ] a 2 b 2 and [R 1/2 U (t)R 1/2 ] a 3 b 3 , by (3.2
.18), we have

|D pq D b jk [Y T U (t)R 1/2 ] kj | ≤ C b+1 (t) n (1+b)/2 α (Y αk ) 2 (b+1)/2 α (Y αq ) 2 1/2 . If q = k, D pk D b jk [U (t)Y ] jk =iD b jk ([R 1/2 U (t)R 1/2 ] jp * [Y T U (t)Y ] kk + [Y T U (t)R 1/2 ] qj * [Y T U (t)R 1/2 ] kp + [R 1/2 U (t)R 1/2 ] jp ).
Then we get

|D pq D b jk [R 1/2 U (t)R 1/2 ] kj | ≤ C b+1 (t) n (1+b)/2 α (Y αk ) 2 (b+2)/2
.

We obtain that

|D pq D b jk [Y T U (t)R 1/2 ] kj | ≤ C b+1 (t) n (1+b)/2 α (Y αk ) 2 (b+1)/2 α (Y αq ) 2 1/2
.

With the same method, we can easily extend the result for all a ∈ N,

|D a pq D b jk [Y T U (t)R 1/2 ] kj | ≤ C a+b (t) n (a+b)/2 α (Y αk ) 2 (b+1)/2 α (Y αq ) 2 a/2 . (3.5.2)

Proof of Proposition 3.2.5

We begin with (3.2.26),

E jk [R 1/2 U (t 1 , s)R 1/2 ] jj [Y T U (t 2 , s)R 1/2 ] kj Z ≤ E 1/2 jk [R 1/2 U (t 1 , s)R 1/2 ] jj [Y T s U (t 2 , s)R 1/2 ] kj 2 K Z ≤ E 1/2 j [R 1/2 U (t 1 , s)R 1/2 ] jj k [Y T s U (t 2 , s)R 1/2 ] kj 2 K Z ≤ E 1/2 j [R 1/2 U (t 1 , s)R 1/2 ] jj 2 E 1/2 j k [Y T s U (t 2 , s)R 1/2 ] kj 2 K Z ≤ C 0 √ nE 1/2   j k [Y T U (t 2 , s)R 1/2 ] kj 2   K Z ≤ C 0 √ nE 1/2   j k,α [R 1/2 U (t 2 , s)] jα Y αk 2   K Z ≤ C 0 √ nE 1/2   k 1 ,α 1 ,k 2 ,α 2   j [R 1/2 U (t 2 , s)] jα 1 [R 1/2 Ū (t 2 , s)] jα 2   |[Y s ] α 1 k 1 [Y s ] α 2 k 2 |   K Z ≤ C 0 √ nE 1/2   k 1 ,α 1 ,k 2 ,α 2 R α 1 α 2 |[Y s ] α 1 k 1 [Y s ] α 2 k 2 |   K Z ≤ C 0 √ nE 1/2   k 1 ,α 1 ,k 2 |[Y s ] α 1 k 1 [Y s ] α 1 k 2 |   K Z ≤ C 0 n 3/2 K Z .
Now we estimate (3.2.28).

E j,k [Y T s U (t 1 , s)R 1/2 ] kj [Y T s U (t 2 , s)R 1/2 ] kj [Y T s U (t 3 , s)R 1/2 ] kj Z ≤ E 1/2 j,k [Y T s U (t 1 , s)R 1/2 ] kj [Y T s U (t 2 , s)R 1/2 ] kj [Y T s U (t 3 , s)R 1/2 ] kj 2 K Z ≤ E 1/2 α 1 ,α 2 α 3   j [R 1/2 U (t 1 , s)] jα 1 [R 1/2 U (t 2 , s)] jα 2 [R 1/2 U (t 3 , s)] jα 3   × k [Y s ] α 1 k [Y s ] α 2 k [Y s ] α 3 k 2 K Z ≤ E 1/2   α 1 α 2 α 3 j [R 1/2 U (t 1 , s)] jα 1 [R 1/2 U (t 2 , s)] jα 2 [R 1/2 U (t 3 , s)] jα 3 2   ×E 1/2   α 1 α 2 α 3 k [Y s ] α 1 k [Y s ] α 2 k [Y s ] α 3 k 2   K Z , E 1/2   α 1 α 2 α 3 j [R 1/2 U (t 1 , s)] jα 1 [R 1/2 U (t 2 , s)] jα 2 [R 1/2 U (t 3 , s)] jα 3 2   = E 1/2   α 1 α 2 α 3 j 1 ,j 2 [R 1/2 U (t 1 , s)] j 1 α 1 [R 1/2 U (t 2 , s)] j 1 α 2 [R 1/2 U (t 3 , s)] j 1 α 3 ×[R 1/2 Ū (t 1 , s)] j 2 α 1 [R 1/2 Ū (t 2 , s)] j 2 α 2 [R 1/2 Ū (t 3 , s)] j 2 α 3 = E 1/2   j 1 ,j 2 α 1 [R 1/2 U (t 1 , s)] j 1 α 1 [R 1/2 Ū (t 1 , s)] j 2 α 1 α 2 [R 1/2 U (t 2 , s)] j 1 α 2 [R 1/2 Ū (t 2 , s)] j 2 α 2 × α 3 [R 1/2 U (t 3 , s)] j 1 α 3 [R 1/2 Ū (t 3 , s)] j 2 α 3 = E 1/2   j 1 ,j 2 |R j 1 j 2 | 3   ≤ C 0 E 1/2   j 1 ,j 2 |R j 1 j 2 | 2   ≤ C 0 √ n, (3.5.3)
and

E 1/2   α 1 α 2 α 3 k [Y s ] α 1 k [Y s ] α 2 k [Y s ] α 3 k 2   = E 1/2   α 1 α 2 α 3 k 1 ,k 2 [Y s ] α 1 k 1 [Y s ] α 2 k 1 [Y s ] α 3 k 1 [Y s ] α 1 k 2 [Y s ] α 2 k 2 [Y s ] α 3 k 2   = E 1/2   k 1 ,k 2 [Y T s Y s ] k 2 k 1 [Y T s Y s ] k 2 k 1 [Y T s Y s ] k 2 k 1   ≤ C 0 √ nE 1/2   k 1 ,k 2 [Y T s Y s ] 2 k 2 k 1   ≤ C 0 √ nE 1/2 Tr[Y T s Y s Y T s Y s ] ≤ C 0 n. (3.5.4)
(3.5.3) and (3.5.4) imply (3.2.28). Same method applies for (3.2.27) :

E jk [Y T s U (t 1 , s)Y s ] kk [Y T s U (t 2 , s)R 1/2 ] kj [R 1/2 U (t 3 , s)R 1/2 ] jj Z ≤ E 1/2 k [Y T s U (t 1 , s)Y s ] kk   j [Y T s U (t 2 , s)R 1/2 ] kj [R 1/2 U (t 3 , s)R 1/2 ] jj   2 K Z ≤ C 0 E 1/2 k [Y T s U (t 1 , s)Y s ] kk 2 E 1/2   k j [Y T s U (t 2 , s)R 1/2 ] kj [R 1/2 U (t 3 , s)R 1/2 ] jj 2   K Z ≤ C 0 √ nE 1/2 k [Y T s U (t 1 , s)Y s ] kk 2 E 1/2   j,k [Y T s U (t 2 , s)R 1/2 ] kj 2   K Z ≤ C 0 √ nE 1/2   k α [Y s ] 2 αk 2   E 1/2 Tr(Y T s U (t 2 , s)R Ū (t 2 , s)Y s ) K Z ≤ C 0 nE 1/2 k α 1 ,α 2 [Y s ] 2 α 1 k [Y s ] 2 α 2 k K Z ≤ C 0 n 3/2 K Z .
E α Y 2 αk 3 = E α 1 ,α 2 ,α 3 Y 2 α 1 k Y 2 α 2 k Y 2 α 3 k = E α 1 =α 2 =α 3 Y 6 α 1 k + E α 1 =α 2 =α 3 3Y 4 α 1 k Y 2 α 3 k +E α 1 =α 2 =α 3 3Y 2 α 1 k Y 2 α 2 k Y 2 α 3 k ≤ C 0 E α 1 Y 4 α 1 k + C 0 ≤ C 0 . When ℓ = 4, E α Y 2 αk 4 = E α 1 ,α 2 ,α 3 ,α 4 Y 2 α 1 k Y 2 α 2 k Y 2 α 3 k Y 2 α 4 k = E α 1 Y 8 α 1 k + E α 1 =α 2 =α 3 =α 4 4Y 6 α 1 k Y 2 α 4 k + E α 1 =α 2 =α 3 =α 4 6Y 4 α 1 k Y 2 α 3 k Y 2 α 4 k +E α 1 =α 2 =α 3 =α 4 3Y 4 α 1 k Y 4 α 3 k + E α 1 =α 2 =α 3 =α 4 Y 2 α 1 k Y 2 α 2 k Y 2 α 3 k Y 2 α 4 k ≤ C 0 E α 1 Y 4 α 1 k + C 0 E α 1 =α 4 Y 4 α 1 k Y 2 α 4 k + C 0 ≤ C 0 .

Proof of Lemma 3.4.2

With the derivatives in (3.4.4), we have

∂TrQ(z)A ∂X ij = - 1 √ n [Y * Q(z)AQ(z)R 1/2 ] ji , ∂TrQ(z)A ∂ Xij = - 1 √ n [R 1/2 Q(z)AQ(z)Y * ] ij .
Then the Nash-Poicaré inequality yields that

Var 1 n TrQ(z)A ≤ 1 n i,j E 1 n ∂TrQ(z)A ∂X ij 2 + E 1 n ∂TrQ(z)A ∂ Xij 2 ≤ 1 n 3 i,j E [Y * Q(z)AQ(z)R 1/2 ] ji 2 + E [R 1/2 Q(z)AQ(z)Y * ] ij 2 ≤ 2 n 3 E[TrYY * Q(z)AQ(z)RQ * (z)A * Q * (z)] ≤ 2 A 2 n 3 E[Tr(I + zQ(z))Q(z)RQ * (z)Q * (z)] ≤ A 2 n 2 C 1 (|z|)C 4 (|Imz| -1 ).
3.5.7 Proof of Proposition 3.4.4

We first establish (3.4.24). From the definition of t in (3.1.2), we have

Im t = 1 2i ( t(z) -t(z)) = vIm( z n TrRT) + vImz, By Proposition 3.4.3, |ε(z)| ≤ 1 n C 2 (|z|)C 5 (|Imz| -1 ), |ε(z)| ≤ 1 n C 3 (|z|)C 7 (|Imz| -1 ),
this along with (3.5.11) yield that

Im(E mc (z)) > v ′ Imz - 1 n |ε| > |Imz| 3 64(C 2 + |z| 2 ) 2 - 1 n 2 C 2 (|z|)C 5 (|Imz| -1
).

(3.5.13)

Then we consider the set E 1 defined by

E 1 = z ∈ C + : 1 - 1 n 2 C 6 (|z|)C 8 (|Imz| -1 ) > 1 2 .
For z ∈ E 1 ,

Im(E mc (z)) > |Imz| 3 128(C 2 + |z| 2 ) 2 .
(3.5.14) From (3.5.10),

det(I -D ′ ) = v ′ Imz Im(E mc (z)) + v ′ Im( z n ε) Im(E mc (z)) + 1 n Imε Im(E mc (z)) ≥ v ′ |Imz| 2 -v ′ |z ε| nImE mc - |ε| nImE mc (a) ≥ |Imz| 4 64(C 2 + |z| 2 ) 2 - 128|z|(C 2 + |z| 2 ) 2 C 3 (|z|)C 7 (|Imz| -1 ) n 2 |Imz| 5 - 128(C 2 + |z| 2 ) 2 C 2 (|z|)C 5 (|Imz| -1 ) n 2 |Imz| 3 ≥ |Imz| 4 64(C 2 + |z| 2 ) 2 (1 - 1 n 2 C 12 (|z|)C 16 (|Imz| -1 )).
where (a) uses (3.5.14) (3.5.12), (3.4.12) and v ′ ≤ 1 |Imz| 2 . Then we consider the second set

E 2 = z ∈ C + : 1 - 1 n 2 C 12 (|z|)C 16 (|Imz| -1 ) > 1 2 ,
and the set

E = z ∈ C + : 1 - 2 n 2 C 6 (|z|)C 10 (|Imz| -1 ) - 2 n 2 C 12 (|z|)C 16 (|Imz| -1 ) > 0 ⊂ E 1 ∩ E 2 ,
or equivalently :

E = z ∈ C + : 1 - 1 n 2 C 12 (|z|)C 16 (|Imz| -1 ) > 0 .
|det(I -D 0 )| ≥ 1 -|z| 2 |v| 1/2 |v ′ | 1/2 |ṽ| 1/2 |ṽ ′ | 1/2 .
As det(I -D) = 1 -|z| 2 vṽ and det(I

-D ′ ) = 1 -|z| 2 v ′ ṽ′ are positive for z ∈ E, |z| 2 vṽ ≤ 1, |z| 2 v ′ ṽ′ ≤ 1, which imply that |z| 2 |v| 1/2 |v ′ | 1/2 |ṽ| 1/2 |ṽ ′ | 1/2 ≤ 1.
By a direct calculus, (3.4.26) holds true.

luate the variance of these estimators for not-too-large M, N . The purpose of this article is to study the asymptotic fluctuations of the population eigenvalue estimator of [START_REF] Mestre | Improved estimation of eigenvalues of covariance matrices and their associated subspaces using their sample estimates[END_REF] in the case of structured population covariance matrices. A central limit theorem (CLT) is provided to describe the asymptotic fluctuations of the estimators with exact expression for the variance as M, N tend to infinity. An empirical, asymptotically accurate, approximation is also derived. For an application of these results in a cognitive radio context, see for instance [START_REF] Yao | Clt for eigen-inference methods in cognitive radios[END_REF]. The remainder of the section is structured as follows : In Section 4.1.1, the system model is introduced and the main results from [START_REF] Mestre | On the asymptotic behavior of the sample estimates of eigenvalues and eigenvectors of covariance matrices[END_REF][START_REF] Mestre | Improved estimation of eigenvalues of covariance matrices and their associated subspaces using their sample estimates[END_REF] are recalled. In Section 4.1.4, the CLT for the estimator in [START_REF] Mestre | Improved estimation of eigenvalues of covariance matrices and their associated subspaces using their sample estimates[END_REF] is stated and the asymptotic variance derived. In Section 4.1.5, an empirical approximation for the variance is derived. Some simulation results are shown in this section.Technical proofs are postponed to the appendix.

Matrix Model

Consider an N × M matrix X N = (X ij ) whose entries are i.i.d. random variables, with distribution CN (0, 1), i.e. X ij = U +iV , where U, V are both i.i.d. real Gaussian random variables N (0, 1 2 ). Let R N be an N × N Hermitian matrix with L (L being fixed) distinct eigenvalues

ρ 1 < • • • < ρ L with respective multiplicities N 1 , • • • , N L (so that L i=1 N i = N ). Consider now Y N = R 1/2 N X N .
The matrix

Y N = [y 1 , • • • , y M ] is the concatenation of M independent observations [y 1 , • • • , y M ],
where each observation writes

y i = R 1/2 N x i with X N = [x 1 , • • • , x M ].
In particular, the (population) covariance matrix of each observation y i is R N = E y i y H i . In this article, we are interested in recovering information on R N based on the observation

RN = 1 M R 1/2 N X N X H N R 1/2 N ,
commonly referred to as the sample covariance matrix of the y i 's.

It is in general a complicated task to infer the spectral properties of R N based on RN for all finite N, M . Instead, in the following, we assume that N and M are large, and consider the following asymptotic regime :

Assumption A1. The dimensions N, M and (N i ) 1≤i≤L satisfy the following conditions : when

N, M, N i → ∞ , N M → c ∈ (0, ∞) and N i M → c i ∈ (0, ∞) , 1 ≤ i ≤ L. (4.1.1)
This assumption will be shortly referred to as N, M → ∞. Assumption A2. The limiting support S of the eigenvalue distribution of RN is formed of L compact disjoint subsets (S k ; 1 ≤ k ≤ L), often referred to as clusters in the sequel.

From [START_REF] Mestre | Improved estimation of eigenvalues of covariance matrices and their associated subspaces using their sample estimates[END_REF], one can also reformulate this condition in mathematical terms : The limiting support of RN is formed of L disjoint clusters if and only if for

1 ≤ i ≤ L, inf N { M N -Ψ N (i)} > 0, where Ψ N (i) is defined in (4.1.2) and α 1 ≤ • • • ≤ α L-1 are the L-1 distinct real ordered solutions Ψ N (i) =            1 N L r=1 N r ρr ρr-α 1 2 m = 1, max 1 N L r=1 N r ρr ρr-α m-1 2 , 1 N L r=1 N r ρr ρr-αm 2 1 < m < L, 1 N L r=1 N r ρr ρr-α L-1 2 m = L (4.1.2)
of the equation :

1 N L r=1 N r ρ 2 r (ρ r -x) 3 = 0.
This condition is also called the separability condition.

Mestre's Estimator of the population eigenvalues

In [START_REF] Mestre | Improved estimation of eigenvalues of covariance matrices and their associated subspaces using their sample estimates[END_REF], an estimator of the population eigenvalues (ρ k ; 1 ≤ k ≤ L) based on the observations RN is proposed. In the case where F Z is the spectral distribution associated to a Hermitian matrix Z ∈ C N ×N with eigenvalues (ξ i ; 1 ≤ i ≤ N ), the Stieltjes transform m Z of F Z takes the particular form :

m Z (z) = F Z (d λ) λ -z = 1 N N i=1 1 ξ i -z = 1 N Tr (Z -zI N ) -1 ,
which is the normalized trace of the resolvent (Z -zI N ) -1 . Since the seminal paper of Marčenko and Pastur [START_REF] Marcenko | Distributions of eigenvalues for some sets of random matrices[END_REF], the Stieltjes transform has proved to be extremely efficient to describe the limiting spectrum of large dimensional random matrices.

In the following, we recall some elements of the proof of Theorem 4.1.1, necessary for the remainder of the chapter. The following important result is due to Bai and Silverstein [START_REF] Silverstein | On the empirical distribution of eigenvalues of a class of large dimensional random matrices[END_REF] (see also [START_REF] Marcenko | Distributions of eigenvalues for some sets of random matrices[END_REF]). Then, the spectral distribution F RN of the sample covariance matrix RN converges (weakly and almost surely) to a probability distribution F as M, N → ∞, whose Stieltjes transform m(z) satisfies :

m(z) = 1 c m(z) -1 - 1 c 1 z , (4.1.6)
for z ∈ C + = {z ∈ C, ℑ(z) > 0} and where m(z) is defined as the unique solution in C + of :

m(z) = -z -c t 1 + tm(z) F R (dt) -1
.

Note that m(z) is also the Stieltjes transform of a probability distribution F , which turns out to be the limiting spectral distribution of F RN where RN is defined as : . This fact will be of importance in the sequel.

RN 1 M X H N R N X N .
Denote by m N (z) and m N (z) the finite-dimensional counterparts of m(z) and m(z), respectively, defined by the relations :

m N (z) = -z - N M t 1 + tm N (z) F R N (dt) -1 , (4.1.7) m N (z) = M N m N (z) -1 - M N 1 z ,
where m N (z) is the unique solution of (4. With these notations at hand, we can now provide some elements of the proof of Theorem 4.1.1.

Proof :[Elements of proof for Theorem 4.1.1] By Cauchy's formula, write :

ρ k = N N k 1 2iπ Γ k 1 N L r=1 N r w ρ r -w dw ,
where Γ k is a negatively oriented contour taking values on C \ {ρ 1 , • • • , ρ L } and only enclosing ρ k . With the change of variable w = -1 m M (z) , the condition that the limiting support S of the eigenvalue distribution of R N is formed of L distinct clusters (S k , 1 ≤ k ≤ L) (cf. Assumption A2), and standard properties of contour integrals, we can write :

ρ k = M 2iπN k R k z m ′ N (z) m N (z) dz , 1 ≤ k ≤ L (4.1.8)
where R k denotes a negatively oriented, rectangular and symmetric with respect to the abscissa axis, contour which only encloses the corresponding cluster S k . Defining

ρk M 2πiN k R k z m ′ RN (z) m RN (z) dz , 1 ≤ k ≤ L , (4.1.9)
dominated convergence arguments ensure that ρ kρk → 0, almost surely. The integral form of ρk can then be explicitly computed thanks to residue calculus, and this finally yields (4.2.1).

Remark 4.1.2 (About the contour integrals) If R ′ k is another (rectangular and symmetric with respect to the abscissa axis) contour which only encloses the k-th cluster, then the value of the contour integrals in (4.1.8) and (4.1.9) remains unchanged. In particular, we can arbitrarily choose two non-overlapping contours R k and R ′ k of the same cluster S k in the sequel.

The main objective of this chapter is to study the performance of the estimators (ρ k , 1 ≤ k ≤ L). More precisely, we will establish a CLT for (M (ρ kρ k ), 1 ≤ k ≤ L) as M, N → ∞, explicitly characterize the limiting covariance matrix Θ = (Θ kℓ ) 1≤k,ℓ≤L , and finally provide an estimator for Θ.

Fluctuations of the population eigenvalue estimators

The Central Limit Theorem

The main result of this chapter is the following CLT which expresses the fluctuations of (ρ k , 1 ≤ k ≤ L).

(n(ρ k -ρ k ), 1 ≤ k ≤ L) D ------→ M,N →∞ x ∼ N L (0, Θ) ,
where Θ is an L × L matrix whose entries Θ kℓ are given by

Θ kℓ = - 1 4π 2 c k c ℓ R k R ′ ℓ m ′ (z 1 )m ′ (z 2 ) (m(z 1 ) -m(z 2 )) 2 - 1 (z 1 -z 2 ) 2 1 m(z 1 )m(z 2 ) dz 1 dz 2 . (4.1.10)
The contours in (4.1.10) are defined as follows. The contours (R k ; 1 ≤ k ≤ L) and (R ′ k ; 1 ≤ k ≤ L) are negatively oriented rectangles, symmetric with respect to the abscissa axis, and only enclosing the cluster S k . They also verify :

R k ∩ R ℓ = R ′ k ∩ R ′ ℓ = ∅ for k = ℓ , R k ∩ R ′ ℓ = ∅ for all k, ℓ.
In particular, the families (R k ) and (R ′ k ) are non-overlapping. We first outline the main steps of the proof and then provide the details.

Using the integral representation of ρk and ρ k , we get :

M (ρ k -ρ k ) = M 2 2πiN k R k z m ′ RN (z) m RN (z) - m ′ N (z) m N (z) dz .
Let K be the union of the R k 's and the R ′ k 's ; denote by C(K, X) the set of continuous functions from K to a Banach space X endowed with the supremum norm u ∞ = sup K |u|. Consider the process :

(X N , X ′ N , u N , u ′ N ) : K → C 4 z → (X N (z), X ′ N (z), u N (z), u ′ N (z)) where X N (z) = M m RN (z) -m N (z) , X ′ N (z) = M m ′ RN (z) -m ′ N (z) , u N (z) = m RN (z) , u ′ N (z) = m ′ RN (z) .
Then from [START_REF] Bai | No Eigenvalues Outside the Support of the Limiting Spectral Distribution of Large Dimensional Sample Covariance Matrices[END_REF] (see also Proposition 4.1.1), (X N , X ′ N , u N , u ′ N ) almost surely belongs to C(K, C 4 ) for N, M large enough and M (ρ kρ k ) writes :

M (ρ k -ρ k ) = M 2πiN k R k z m N (z)X ′ N (z) -u ′ N (z)X N (z) m N (z)u N (z) dz △ = Υ N (X N , X ′ N , u N , u ′ N ) ,
2. Let Assumptions A1 and A2 hold true and consider a compact set K ⊂ C, symmetric with respect to the real axis (i.e. z ∈ K ⇒ z ∈ K) which does not intersect S. Then, the process

(X N , X ′ N ) : K → C 2 z → (X N (z), X ′ N (z))
converges in distribution to a stochastic process (X, Y ) satisfying X(z) = X(z) and Y (z) = Y (z). As an R 4 -valued real process, the process (X, Y ) is a centered Gaussian process with mean function zero and covariance function defined as follows, for z, z ∈ K : ------→ N,M →∞ (m, m ′ ), a straightforward corollary of Lemma 4.1.1 yields the convergence in distribution of (X N , X ′ N , u N , u ′ N ) to (X, Y, m, m ′ ). This concludes the proof of step (i).

E X(z)X(z) = κ(z, z) , (4.1.12) E Y (z)X(z) = ∂κ ∂z (z, z) , E X(z)Y (z) = ∂κ ∂ z (z, z) , E Y (z)Y (z) = ∂ 2 ∂z∂ z κ(z, z) .
Consider two families of contours (R k ) and (R ′ k ) as described in Theorem 4.2.1. Denote by

K = k=1:L R k ∪ k=1:L R ′ k . (4.1.13)
A direct consequence of Lemma 4.1.1 yields that (X N , X ′ N , u N , u ′ N ) : K → C 4 converges in distribution to the Gaussian process (X, Y, m, m ′ ) with mean (0, 0, m, m ′ ) and covariance structure inherited from the Gaussian process (X, Y ). We are now in position to transfer the convergence of (X N , X ′ N , u N , u ′ N ) to Υ N (X N , X ′ N , u N , u ′ N ) via the continuous mapping theorem, whose statement is reminded below. Theorem 4. 1.4 ([50,Th. 4.27]) For any metric spaces S 1 and S 2 , let ξ, (ξ n ) n≥1 be random elements in S 1 with ξ n D ---→ n→∞ ξ and consider some measurable mappings f , (f n ) n≥1 : S 1 → S 2 and a measurable set Γ ⊂ S 1 with ξ ∈ Γ a.s. such that

f n (s n ) → f (s) as s n → s ∈ Γ. Then f n (ξ n ) D ---→ n→∞ f (ξ).
It remains to apply Theorem 4.2.5 to the process (X N , X ′ N , u N , u ′ N ) and to the function Υ N as defined in (4.1.11). Denote by 2

Υ(x, y, v, w) = 1 2πic k R k z m(z)y(z) -w(z)x(z) m(z)v(z) dz ,
and consider the set

Γ = (x, y, v, w) ∈ C 4 (K, C), inf K
|v| > 0, and x, y, w are continuous over K .

It is obvious that X, Y and m ′ are continuous over K. Then, it is shown in [5, Section 9.12.1] that inf K |m| > 0, and, by a dominated convergence theorem argument, that (x N , y N , v N , w N ) → (x, y, v, w) ∈ Γ implies that Υ N (x N , y N , v N , w N ) → Υ(x, y, v, w). Therefore, Theorem 4.2.5 applies to Υ N (x N , y N , v N , w N ) and the following convergence holds true :

Υ N (X N , X ′ N , u N , u ′ N ) D ------→ M,N →∞ Υ(X, Y, m, m ′ ) ,
and step (ii) is established. It now remains to prove step (iii), i.e. to check the Gaussianity of the random variable Υ(X, Y, m, m ′ ) and to compute the covariance between Υ(X, Y, m, m ′ , C k ) and Υ(X, Y, m, m ′ , C ℓ ).

In order to propagate the Gaussianity of the deviations in the integrands of (4.1.9) to the fluctuations of the integral which defines ρk , it suffices to notice that the integral can be written as the limit of a finite Riemann sum and that a finite Riemann sum of Gaussian random variables is still Gaussian. Therefore M (ρ kρ k ) converges to a real Gaussian distribution (notice that Υ(X, Y, m, m ′ , R k ), being the limiting distribution of the real random variable Υ N , is real as well). The same argument applies to the whole vector (n(ρ kρ k ); 1 ≤ k ≤ L), which hence converges toward a Gaussian vector Υ ∼ N L (m, Θ) :

    M (ρ 1 -ρ 1 )
. . .

M (ρ L -ρ L )     D ------→ M,N →∞ Υ △ =     Υ(X, Y, m, m ′ , R 1 ) . . . Υ(X, Y, m, m ′ , R L )     ,
where m is a L × 1 vector and Θ = (Θ kℓ ) is a L × L covariance matrix.

2. As previously, we shall explicitly indicate the dependence on the contour R k if needed and write Υ(x, x ′ , u, u ′ , R k ).

As inf z∈K |m(z)| > 0, a straightforward application of Fubini's theorem together with the fact that E(X) = E(Y ) = 0 yields :

E z m ′ (z)X(z) m 2 (z) -z Y (z) m(z) dz = 0 , hence m = 0.
It remains to compute the covariance between Υ(X, Y, m, m ′ , R k ) and Υ(X, Y, m, m ′ , R ℓ ). As Θ = E(ΥΥ T ), write :

Θ kℓ = E Υ(X, Y, m, m ′ , R k )Υ(X, Y, m, m ′ , R ℓ ) , (a) = E Υ(X, Y, m, m ′ , R k )Υ(X, Y, m, m ′ , R ′ ℓ ) , = - 1 4π 2 c k c l E R k z 1 m ′ (z 1 )X(z 1 ) m 2 (z 1 ) - Y (z 1 ) m(z 1 ) dz 1 × R ′ ℓ z 2 m ′ (z 2 )X(z 2 ) m 2 (z 2 ) - Y (z 2 ) m(z 2 ) dz 2 ,
where (a) follows from Remark 4.1.2 and enforces the fact that the contours are non-overlapping.

Choosing non-overlapping contours will help us to compute the Θ kℓ 's by evaluating contour integrals with no singularities on the contours. Write :

Θ kℓ = E Υ(X, Y, m, m ′ , R k )Υ(X, Y, m, m ′ , R ′ ℓ ) , (a)
= -

1 4π 2 c k c l R k R ′ ℓ z 1 z 2 m ′ (z 1 )m ′ (z 2 )κ(z 1 , z 2 ) m 2 (z 1 )m 2 (z 2 ) - m ′ (z 1 )∂ 2 κ(z 1 , z 2 ) m 2 (z 1 )m(z 2 ) - m ′ (z 2 )∂ 1 κ(z 1 , z 2 ) m(z 1 )m 2 (z 2 ) + ∂ 2 12 κ(z 1 , z 2 ) m(z 1 )m(z 2 ) dz 1 dz 2 ,
where (a) follows from Lemma 4.1.1 and the fact that inf z∈K |m(z)| > 0 together with Fubini's theorem, and ∂ 1 , ∂ 2 , ∂ 2 12 respectively stand for ∂/∂z 1 , ∂/∂z 2 and ∂ 2 /∂z 1 ∂z 2 . The above double integral is also well-defined as κ(z 1 , z 2 ) is well-defined and C ∞ -differentiable over R k × R ′ ℓ . By integration by parts, we obtain

z 1 z 2 m ′ (z 2 )∂ 1 κ(z 1 , z 2 ) m(z 1 )m 2 (z 2 ) dz 1 = - z 2 m ′ (z 2 )κ(z 1 , z 2 ) m(z 1 )m 2 (z 2 ) + z 1 z 2 m ′ (z 1 )m ′ (z 2 )κ(z 1 , z 2 ) m 2 (z 1 )m 2 (z 2 ) dz 1 .
Similarly,

z 1 z 2 m(z 2 )∂ 12 κ(z 1 , z 2 ) m(z 1 )m 2 (z 2 ) dz 1 = - z 2 ∂ 2 κ(z 1 , z 2 ) m(z 1 )m(z 2 ) dz 1 + z 1 z 2 m ′ (z 1 )∂ 2 κ(z 1 , z 2 ) m 2 (z 1 )m(z 2 ) dz 1 .
Hence

Θ kℓ = - 1 4π 2 c k c l R k R ′ ℓ z 2 m ′ (z 2 )κ(z 1 , z 2 ) m(z 1 )m 2 (z 2 ) dz 1 dz 2 - R k R ′ ℓ z 2 ∂ 2 κ(z 1 , z 2 ) m(z 1 )m(z 2 ) dz 1 dz 2 .
Another integration by parts yields

z 2 ∂ 2 κ(z 1 , z 2 ) m(z 1 )m(z 2 ) dz 2 = - κ(z 1 , z 2 ) m(z 1 )m(z 2 ) dz 2 + z 2 m ′ (z 2 )κ(z 1 , z 2 ) m(z 1 )m 2 (z 2 ) dz 2 .
Finally, we obtain :

Θ kℓ = - 1 4π 2 c k c ℓ R k R ′ ℓ κ(z 1 , z 2 ) m(z 1 )m(z 2 ) d z 1 d z 2 ,
and (4.1.10) is established. We first address the convergence of Θkℓ to Θ kℓ . Due to [START_REF] Bai | No Eigenvalues Outside the Support of the Limiting Spectral Distribution of Large Dimensional Sample Covariance Matrices[END_REF][START_REF] Bai | Exact Separation of Eigenvalues of Large Dimensional Sample Covariance Matrices[END_REF], almost surely, the eigenvalues of RN will eventually belong to any ε-blow-up of the support S of the probability measure associated to m, i.e. the set {x ∈ R : d(x, S) < ε}. Hence, if ε is small enough, the distance between these eigenvalues and any z ∈ K will be eventually uniformly lower-bounded. By [64, Lemma 1], the same holds true for the zeros of m RN (which are real). In particular, this implies that m RN is eventually uniformly lower-bounded on K (if not, then by compacity, there would exist z ∈ K such that m RN (z) = 0 which yields a contradiction because all the zeroes of m RN are strictly within any contour). With these arguments at hand, one can easily apply the dominated convergence theorem and conclude that Θkℓ → Θ kℓ a.s.

Estimation of the covariance matrix

Θkℓ = M 2 N k N ℓ   (i,j)∈N k ×N ℓ , i =j -1 (μ i -μj ) 2 m ′ RN (μ i )m ′ RN (μ j ) +δ kℓ i∈N k   m ′′′ RN (μ i ) 6m ′ RN (μ i ) 3 - m ′′ RN (μ i ) 2 4m ′ RN (μ i ) 4
= - M 2 4π 2 N k N ℓ R k R ′ ℓ m ′ RN (z 1 )m ′ RN (z 2 ) (m RN (z 1 ) -m RN (z 2 )) 2 - 1 (z 1 -z 2 ) 2 × 1 m RN (z 1 )m RN (z 2 ) d z 1 d z 2 .
We now evaluate the integral (4.1.15) by computing the residues of the integrand within R k and R ′ ℓ . There are two cases to discuss depending on whether k = ℓ and k = ℓ. Denote by h(z 1 , z 2 ) the integrand in (4.1.15), that is :

h(z 1 , z 2 ) = m ′ RN (z 1 )m ′ RN (z 2 ) (m RN (z 1 ) -m RN (z 2 )) 2 - 1 (z 1 -z 2 ) 2 × 1 m RN (z 1 )m RN (z 2 ) . (4.1.16) Note that, when z 2 is fixed, for z 1 → λi , m ′ RN (z 1 ) (m RN (z 1 ) -m RN (z 2 )) 2 a.s. ----→ z 1 → λi M. Then h(z 1 , z 2 )
a.s.

--→ 0 when z 1 → λi . Same result holds for z 1 → 0. That is to say, λi and 0 are not poles of h(z 1 , z 2 ).

To apply the residue theorem, we first consider the case where k = ℓ. In this case, the two integration contours are different and never intersect (in particular, z 1 is always different from z 2 ). Let z 2 be fixed, and denote by μi the zeros (labeled in increasing order) of m RN , then the computation of the residue Res(h(•, z 2 ), μi ) of h(•, z 2 ) at a zero μi of m RN which is located within R k is straightforward and yields

r(z 2 ) △ = Res(h(•, z 2 ), μi ) = m ′ RN (μ i ) m ′ RN (z 2 ) m 2 RN (z 2 ) - 1 (μ i -z 2 ) 2 1 m ′ RN (μ i )m RN (z 2 ) . (4.1.17)
Similarly, if one computes Res(r, μj ) at a zero μj of m RN located within R ′ ℓ , one obtains :

Res(r, μj ) = - 1 (μ i -μj ) 2 m ′ RN (μ i )m ′ RN (μ j ) .
As stated in the following proposition, let

z 2 ∈ R ′ ℓ \R, the set R z 2 = {z 1 ∈ C : z 1 = z 2 , m RN (z 1 ) = m RN (z 2 ) = 0}
is eventually empty a.s. for all N, M large, and if z 2 ∈ R ′ ℓ ∩ R, this set is not empty, however, the integration with respect to z 2 for this residue is zero because the set R ′ ℓ ∩ R only contains two points, hence the residue in this set has not to be counted.

R z 2 = {z 1 ∈ C : z 1 = z 2 , m RN (z 1 ) = m RN (z 2 ) = 0} = ∅ for all N, n large.
The proof of Proposition 4.1.2 is postponed to Appendix 4.3.3. It remains to count the number of μi within each contour. By [64, Lemma 1], eventually, there are exactly as many μi as eigenvalues within each contour, hence the result in the case k = ℓ :

Θkℓ = M 2 N k N ℓ (i,j)∈N k ×N ℓ - 1 (μ i -μj ) 2 m ′ RN (µ i )m ′ RN (µ j )
.

We now compute the integral (4.1.15) in the case where k = ℓ, and begin by the computation of the residues at μi . The definition (4.1.17) of r and the computation of Res(r, μj ) still hold true in the case where μj is within R k but different from μi . It remains to compute Res(r, μi ). Taking z 2 → µ i , we get :

lim z 2 →μ i (z 2 -μi ) 3 1 m ′ RN (μ i )m RN (z 2 )(μ i -z 2 ) 2 = 1 m ′ 2 RN (μ i ) , lim z 2 →μ i (z 2 -μi ) 2   1 m ′ RN (μ i )m RN (z 2 )(μ i -z 2 ) 2 1 m ′ RN 2 (μ i )(z 2 -μi ) 3   = - m ′′ RN (μ i ) 2m ′ RN 3 (μ i ) .
Finally,

lim z 2 →μ i (z 2 -μi )   1 m ′ RN (μ i )m RN (z 2 )(μ i -z 2 ) 2 - 1 m ′ 2 RN (μ i )(z 2 -μi ) 3 + m ′′ R N (μ i ) 2m ′ 3 R N (μ i )(z 2 -μi ) 2   = m ′′′ RN (μ i ) 6m ′ RN (μ i ) 3 - m ′′ RN (μ i ) 2 4m ′ RN (μ i ) 4 .
Hence the residue :

Res(r, μi ) = m ′′′ R N (μ i ) 6m ′ R N (μ i ) 3 - m ′′ R N (μ i ) 2 4m ′ R N (μ i ) 4 .
There are two other cases that should be taken into account for the computation of the integral :

The set R z 2 , and the residue for z 1 = z 2 . The first case can be handled as before. For

z 1 = z 2 , note that m ′ RN (z 1 )m ′ RN (z 2 ) (m RN (z 1 ) -m RN (z 2 )) 2
a.s.

----→

z 1 →z 2 1.
z 1 = z 2 is not the residue for this term. It remains to compute

1 (z 1 -z 2 ) 2 1 m RN (z 1 )m RN (z 2 )
for the residue z 1 = z 2 . The integration by parts formula yields that :

1 (z 1 -z 2 ) 2 dz 1 m RN (z 1 )m RN (z 2 ) = - m ′ RN (z 1 ) (z 1 -z 2 ) dz 1 m 2 RN (z 1 )m RN (z 2 ) .
Then the residue for z 1 = z 2 is :

- m ′ RN (z 2 ) m 3 RN (z 2 ) .
This is the derivative function of

1 2m 2 RN (z 2 )
, then the integration with respect to z 2 is zero. Finally both have a null contribution, hence the formula :

Θkk = M 2 N 2 k   (i,j)∈N 2 k , i =j -1 (μ i -μj ) 2 m ′ RN (μ i )m ′ RN (μ j ) + i∈N k   m ′′′ RN (μ i ) 6m ′ RN (μ i ) 3 - m ′′ RN (μ i ) 2 4m ′ RN (μ i ) 4     .

Performance of cognitive radios

We consider the system model introduced in Section 4.1.1. Assuming the spectrum of R M allows one to clearly distinguish the successive clusters (as in Figure 4.1), Proposition 4.1.1 enables the detection of primary transmitters and the estimation of their transmit powers P 1 , . . . , P K ; this boils down to estimating the largest K eigenvalues of WPW H + σ 2 I N , i.e. the P k + σ 2 , and to subtract σ 2 (optionally estimated from the smallest eigenvalue of WPW H + σ 2 I N if n < N ). Call Pk the estimate of P k .

Based on these power estimates, the sensor can determine the optimal coverage for secondary communications that ensures no interference to the primary network. A basic idea for instance is to ensure that the closest primary user, i.e. that with strongest received power, is not interfered. Our interest is then cast on P K . Now, since the power estimator is imperfect, it is hazardous for the secondary network to state that K has power PK or to add some empirical security margin to PK . The results of Section 4.1.4 partially answer this problem.

Theorems 4.1.3 and 4.1.5 enable the secondary sensor to evaluate the accuracy of Pk . In particular, assume that the cognitive radio protocol allows the secondary network to interfere the primary network with probability q and denote A the value

A ∆ = inf a {Pr(P K -PK > a) ≤ q}.
According to Theorem 4.1.3, for N, M large, A is well approximated by ΘK,K Q -1 (q), with Q the Gaussian Q-function. If the sensor detects a user with power P K , estimated by PK , Pr( PK + A < P K ) < q and then it is safe for the secondary network to assume the worst case scenario where user K transmits at power PK + A ≃ PK + ΘK,K Q -1 (q).

In Figure 4.3, the performance of Theorem 4.1.3 is compared against 10, 000 Monte Carlo simulations of a scenario of three users, with n 1 = n 2 = n 3 = 20, N = 60 and M = 600. It appears that the limiting distribution is very accurate for these values of N, M . We also performed simulations to obtain empirical estimates Θk,k of Θ k,k from Theorem 4.1.5, which suggest that Θk,k is an accurate estimator as well.

Estimation of Parametrized Covariance Matrices

This section is based on the paper [START_REF] Yao | Eigenvalue estimation of parameterized covariance matrices of large dimensional data[END_REF]. Estimating the covariance matrix of a series of independent multivariate observations is a crucial issue in many signal processing applications. A reliable estimate of the covariance matrix is for instance needed in principal component analysis [START_REF] Jolliffe | Principal Component Analysis[END_REF], direction of arrival estimation for antenna arrays [START_REF] Schmidt | Multiple emitter location and signal parameter estimation[END_REF], blind subspace estimation [START_REF] Moulines | Subspace Methods for the Blind Identification of Multichannel FIR Filters[END_REF], capacity estimation [START_REF] Foschini | On limits of wireless communications in a fading environment when using multiple antennas[END_REF], estimation/detection procedures [START_REF] Schmidt | Multiple emitter location and signal parameter estimation[END_REF][START_REF] Wax | Detection of signals by information theoretic criteria[END_REF], etc.

In the case where the dimension N of the observations is small compared to the number M of observations, the empirical covariance matrix based on the observations often provides a good estimate for the unknown covariance matrix. This estimate becomes however much less accurate, and even not consistent with the dimension N getting higher (see for instance [START_REF] Mestre | On the asymptotic behavior of the sample estimates of eigenvalues and eigenvectors of covariance matrices[END_REF]Theorem 2]). An interesting theoretical framework for modern estimation of multi-dimensional variables occurs whenever the number of available samples M grows at the same pace as the dimension N of the considered variables. Shifting to this new assumption induces fundamental differences in the behavior of the empirical covariance matrix as analyzed in Mestre's work [START_REF] Mestre | On the asymptotic behavior of the sample estimates of eigenvalues and eigenvectors of covariance matrices[END_REF][START_REF] Mestre | Improved estimation of eigenvalues of covariance matrices and their associated subspaces using their sample estimates[END_REF]. Recently, several attempts have been done to address this problem (cf. [START_REF] Mestre | On the asymptotic behavior of the sample estimates of eigenvalues and eigenvectors of covariance matrices[END_REF][START_REF] Mestre | Improved estimation of eigenvalues of covariance matrices and their associated subspaces using their sample estimates[END_REF][START_REF] Karoui | Spectrum estimation for large dimensional covariance matrices using random matrix theory[END_REF][START_REF] Bai | On estimation of the population spectral distribution from a high-dimensional sample covariance matrix[END_REF][START_REF] Kammoun | Performance of capacity inference methods under colored interference[END_REF][START_REF] Ledoit | Nonlinear Shrinkage Estimation of Large-Dimensional Covariance Matrices[END_REF]) using large random matrix theory which proposed powerful tools, mainly spurred by Girko's G-estimators [START_REF] Girko | Ten years of general statistical analysis[END_REF], to cope with this new context. In [START_REF] Mestre | On the asymptotic behavior of the sample estimates of eigenvalues and eigenvectors of covariance matrices[END_REF][START_REF] Mestre | Improved estimation of eigenvalues of covariance matrices and their associated subspaces using their sample estimates[END_REF], Mestre considers the eigenvalue estimation of a parametrized model of covariance matrices similar to the model we shall study in this article. In [START_REF] Karoui | Spectrum estimation for large dimensional covariance matrices using random matrix theory[END_REF] and [START_REF] Ledoit | Nonlinear Shrinkage Estimation of Large-Dimensional Covariance Matrices[END_REF], grid-based techniques for inverting the Marčenko-Pastur equation are proposed. In [START_REF] Kammoun | Performance of capacity inference methods under colored interference[END_REF], the problem of estimating a specific linear functional of the eigenvalues of an unknown covariance matrix is addressed. In [START_REF] Bai | On estimation of the population spectral distribution from a high-dimensional sample covariance matrix[END_REF], the eigenvalues of an unknown parametrized covariance matrix are estimated by resorting on the empirical moments of the observations. This technique, which goes back to Pisarenko's ideas [START_REF] Pisarenko | The retrieval of harmonics from a covariance function[END_REF], will be also combined to large random matrix theory here.

We shall consider the case where the dimension of each observation N together with the number of samples M go to infinity at the same pace, i.e. their ratio converges to some nonnegative constant c > 0. In order to present the contribution provided in this paper, let us describe the model under study.

Consider an N × M matrix X N = (X ij ) whose entries are independent and identically distributed (i.i.d.) random variables. Let R N be an N × N Hermitian matrix with L (L being fixed and known) distinct eigenvalues 0 < ρ

1 < • • • < ρ L with respective multiplicities N 1 , • • • , N L (notice that L i=1 N i = N ). Consider now Y N = R 1/2 N X N . The matrix Y N = [y 1 , • • • , y M ]
is the concatenation of M independent observations, where each observation writes

y i = R 1/2 N x i with X N = [x 1 , • • • , x M ].
In particular, the covariance matrix of each observation y i is R N = Ey i y H i (matrix R N is sometimes called the population covariance matrix).

In this chapter, we consider the problem of estimating individually the eigenvalues ρ i as well as their multiplicities N i in the case where the total number of eigenvalues is fixed and known.

Such a scenario is customary in applications for wireless communications. A relevant example concerns uplink CDMA systems operating over flat fading channels, where users are arranged into L classes, each class corresponding to a distinct power amount. In this case, matrix Y N can be modeled as :

Y N = WP 1 2 X + σV N
where W are V N represent respectively the signature matrix assumed to be orthogonal and the noise matrix, while P is diagonal with diagonal elements taking distinct values among the finite mulations are presented in Section 4.2.4. Finally, the remaining technical details are postponed to the Appendices.

Main assumptions and general background

Consider the model

Y N = R 1/2 N X N , and RN = 1 M Y N Y H N .
At first, an assumption about the matrix R N is needed : Assumption 4.2.1 R N is an N × N Hermitian non-negative definite matrix with L (L being fixed and known) distinct eigenvalues 0

< ρ 1 < • • • < ρ L with respective multiplicities N 1 , • • • , N L (notice that L i=1 N i = N ).
As mentioned earlier, we consider the asymptotic regime where the number of samples M and the dimension N grow to infinity at the same pace, together with the multiplicities of each eigenvalue of R N . Assumption 4.2.2 Let M, N be integers such that :

N, M → ∞ , with N M → c ∈ (0, ∞) , and
N i N → c i ∈ (0, ∞) , 1 ≤ i ≤ L. ( 4

.2.1)

This assumption will be shortly referred to as N, M → ∞.

The following assumption is standard and is sufficient for estimation purposes.

Assumption 4.2.3 Let X N = (X ij ) be a N ×M matrix whose entries are i.

i.d. random variables in C such that E(X 1,1 ) = 0, E(|X 1,1 | 2 ) = 1 with finite fourth moment : E(|X 1,1 | 4 ) < ∞.
Remark 4.2.1 In order to establish the fluctuations of the estimators, the Gaussianity of the entries of X N is needed (although this technical condition may be removed with substantial extra work).

Assumption 3b : The entries of the N × M matrix X N = (X ij ) are i.i.d. standard complex Gaussian variables, i.e. X ij = U + iV , where U, V are both independent real Gaussian random variables N (0, 1 2 ). It is well-known in large random matrix theory that under Assumptions 4.2.1, 4.2.2 and 4.2.3, F RN converges to a limiting probability distribution (cf. Theorem 4.1.2). In Mestre's paper [START_REF] Mestre | Improved estimation of eigenvalues of covariance matrices and their associated subspaces using their sample estimates[END_REF], a separability condition (cf. Section 4.1.1) is needed in order to derive the estimator of R N 's eigenvalues :

(M (ρ k -ρ k ), 1 ≤ k ≤ L) D ------→ M,N →∞ x ∼ N L (0, Θ) ,
where N L refers to a real L-dimensional Gaussian distribution, and Θ is a L × L matrix whose entries Θ kℓ are given by,

Θ kℓ = - 1 4π 2 c 2 c k c ℓ C k C ℓ m ′ (z 1 )m ′ (z 2 ) (m(z 1 ) -m(z 2 )) 2 - 1 (z 1 -z 2 ) 2 1 m(z 1 )m(z 2 ) dz 1 dz 2 ,
where C k (resp. C ℓ ) is a closed counterclockwise oriented contour which only contains the k-th cluster (resp. ℓ-th) .

The main objective of this section is to provide estimators for the ρ k 's without relying any more on the separability condition (i.e. to remove Assumption 4.2.4). A Central Limit Theorem will be established as well for the proposed estimator. As a by-product, the knowledge of the multiplicities will no longer be needed, and they will be estimated as well.

as in Lemma 4.2.1. Consider the following system of equations :

   L i=1 x i = 1, L i=1 x i y k i = γk for 1 ≤ k ≤ 2L -1, (4.2.4)
where (x i ) 1≤i≤L and (y i ) 1≤i≤L are 2L unknown parameters. Then for N, M large enough, the system of equations (4.2.4) has one and only one real solution

(ĉ 1 , • • • , ĉL , ρ1 , • • • , ρL ) with ρ1 < • • • < ρL . Moreover, (ĉ 1 , • • • , ĉL , ρ1 , • • • , ρL ) is a consistent estimator of (c 1 , • • • , c L , ρ 1 , • • • , ρ L ),
i.e., ĉℓc ℓ a.s.

------→ N,M →∞ 0 and ρℓρ ℓ a.s.

------→

N,M →∞ 0, with c ℓ = lim N ℓ N for 1 ≤ ℓ ≤ L. Remark 4.2.3
The condition of separability is not required in the previous theorem. Moreover, the multiplicities are assumed to be unknown and thus have to be estimated. Fig. 4.2 represents a case where the three clusters are merged into one cluster. In such a situation, the estimator in [START_REF] Mestre | Improved estimation of eigenvalues of covariance matrices and their associated subspaces using their sample estimates[END_REF] is biased whereas the proposed one is asymptotically consistent.

Remark 4.2.4

We use the estimator proposed in Lemma 4.2.1. However, the proof below does not depend on the estimator of the moments we choose. In fact, for any consistent estimator of the moments γ i , the above theorem always holds true.

Proof of Theorem 4.2.2

The proof can be split into two main steps. By using the inverse function theorem, we can prove the almost sure existence of a real solution. Then, the uniqueness is ensured by a matrix inversion argument.

Existence of a real solution of the system

The first task is to show that the system of equations (4.2.4) admits, for N sufficiently large, one real solution

(ĉ 1 , • • • , ĉL , ρ1 • • • , ρL ) satisfying ρ1 < ρ2 < • • • < ρL .
We shall also establish the consistency of the obtained solution. The proof of the existence of a real solution follows in the same way as in [START_REF] Bai | On estimation of the population spectral distribution from a high-dimensional sample covariance matrix[END_REF]. It is merely based on the use of the inverse function theorem which ensures the existence as soon as the Jacobian matrix of the considered transformation is invertible. We recall below the inverse function theorem [START_REF] Krantz | Function Theory of Several Complex Variables[END_REF] : Theorem 4. 2.3 ([56]) Let f : R n → R n be a continuously differentiable function. Let a and b be vectors of R n such that f (a) = b. If the Jacobian matrix of f at a is invertible, then there exists a neighborhood U containing a such that f : U → f (U ) is a diffeomorphism, i.e, for every y ∈ f (U ) there exists a unique x such that f (x) = y. In particular, f is invertible in U .

Consider the functional f defined as :

f (x 1 , • • • , x L , y 1 , • • • , y L ) = L ℓ=1 x ℓ , L ℓ=1 x ℓ y ℓ , • • • , L ℓ=1 x ℓ y 2L-1 ℓ . Consider z = (x 1 , • • • , x L , y 1 , • • • , y L ) and denote by c = (c 1 , • • • , c L , ρ 1 , • • • , ρ L ) ; we then have : M ∂f ∂z z=c =       1 • • • 1 0 • • • 0 ρ 1 • • • ρ L c 1 • • • c L . . . . . . . . . . . . . . . . . . ρ 2L-1 1 • • • ρ 2L-1 L (2L -1)c 1 ρ 2L-2 1 • • • (2L -1)c L ρ 2L-2 L       . (4.2.5)
As proven in [4, Proposition 1], matrix M is invertible. The inverse function theorem then applies. Denote by

ψ i = L k=1 c k ρ i k for 0 ≤ i ≤ 2L -1. There exists a neighborhood U of (c 1 , • • • , c L , ρ 1 , • • • , ρ L ) and a neighborhood V of (ψ 0 , • • • , ψ 2L-1 )
such that f is a diffeomorphism from U onto V . On the other hand, we have :

γi -γ i a.s.
--→ 0.

As γ i -ψ i → 0, therefore, almost surely, (γ 0 , • • • , γ2L-1 ) ∈ V for N and M large enough. Hence, a real solution (ĉ 1 , • • • , ĉL , ρ1 , • • • , ρL ) = f -1 (γ 0 , • • • , γ2L-1 ) ∈ U
exists. And by the continuity, one can get easily that :

ĉℓ -c ℓ a.s. ------→ N,M →∞ 0 and ρℓ -ρ ℓ a.s. ------→ N,M →∞ 0 for 1 ≤ ℓ ≤ L .

Uniqueness of the solution of the system

Consider the polynomial Q with degree L defined as :

Q(X) = L ℓ=0 (X -ρℓ ) △ = L ℓ=0 s ℓ X ℓ where s L = 1. Denote by s = [s 0 , • • • , s L-1 ] T . It is clear that g : (ρ 1 , • • • , ρL ) → s is a homeo- morphism.
It remains thus to show that vector s is uniquely determined by (γ 0 , • • • , γ2L-1 ).

It is clear that each ρk is also the zero of the polynomial functions R ℓ (X) given by :

R ℓ (X) = L i=0 s i X i+ℓ ,
where 0 ≤ ℓ ≤ L -1. In other words, for 1 ≤ k ≤ L, we get : 4. Determine (by using for instance function roots of matlab) the roots ρ1 , • • • , ρL of the polynomial whose coefficients are given by vector s.

5.

Construct matrix A as specified by (4.2.10), and vector

d = [γ 0 , • • • , γL-1 ] T . 6. The coefficient estimates ĉ = [ĉ 1 , • • • , ĉL ]
T are thus given by:

ĉ = A -1 d.
Remark 4.2.5 Note that while the existence of a real solution is only proven for M and N large enough, the previous algorithm always yield a solution, even for very small dimensions. However, in such scenarios, the validity of the obtained solution is not ensured. In fact, if N and M are not large enough, the moment estimates are not accurate, and the solution of the algorithm may yield complex or negative eigenvalues. This event completely disappears when N and M or only M take higher values. In practice, getting such inadequate solutions should warn that more samples are required.

Fluctuations of the estimator

In this section, we shall study the fluctuations of the multiplicities and eigenvalues estimators

(ĉ 1 , • • • , ĉL , ρ1 , • • • , ρL ) introduced in Theorem 4.2.2.
In particular, we establish a Central Limit Theorem for the whole vector in the case where the entries of matrix X N are Gaussian.

M ĉ1 - N 1 N , • • • , ĉL - N L N , ρ1 -ρ 1 , • • • , ρL -ρ L D ------→ N,M →∞ N 2L (0, Θ)
where Θ is a 2L × 2L matrix admitting the decomposition Θ = M -1 WM -1 T and matrix M is the Jacobian matrix of f evaluated for z = c and is defined in (4.2.5) and

W = 0 0 0 V ,
where V is a (2L -1) × (2L -1) matrix whose entries are given by (for 1 ≤ k, ℓ ≤ 2L -1) :

V k,ℓ = - (-1) k+ℓ 4π 2 c 2 C 1 C 2 m ′ (z 1 )m ′ (z 2 ) (m(z 1 ) -m(z 2 )) 2 - 1 (z 1 -z 2 ) 2 × 1 m k (z 1 )m ℓ (z 2 ) d z 1 d z 2
where C 1 and C 2 are two closed contours non-overlapping which contain the support S of F and are counterclockwise oriented.

Proof : The proof relies on the same techniques as developed in [START_REF] Yao | Fluctuations of an improved population eigenvalue estimator in sample covariance matrix models[END_REF]. We outline hereafter the main steps and then provide the details.

By Theorem 4.2.2, the estimate vector (ĉ 1 , • • • , ĉL , ρ1 , • • • , ρL ) verifies the following system of equations :

     L i=1 ĉi = 1, L i=1 ĉi ρi = γ1 , L i=1 ĉi ρk i = γk for 2 ≤ k ≤ 2L -1,
where the γi 's are the moment estimates provided by Lemma 4.2.1.

Using the integral representation of L i=1 c i ρ i and L i=1 c i ρ k i (cf. Section 4.3.4 in the Appendix and Formula (4.3.15)), we get :

             L i=1 M ĉi -N i N = 0, L i=1 M ĉi ρi -N i N ρ i = -M 2 2N iπ C z m ′ RN (z) m RN (z) - m ′ N (z) m N (z) dz, L i=1 M ĉi ρk i -N i N ρ k i = M 2 (-1) k 2i(k-1)N π C 1 m RN (z) k-1 - 1 m N (z) k-1 dz, 2 ≤ k ≤ 2L -1.
Denote by C(C, C) the set of continuous functions from C to C endowed with the supremum norm u ∞ = sup C |u|. In the same way as in [START_REF] Yao | Fluctuations of an improved population eigenvalue estimator in sample covariance matrix models[END_REF], consider the process :

(X N , X ′ N , u N , u ′ N ) : C → C, where X N (z) = M m RN (z) -m N (z) , X ′ N (z) = M m ′ RN (z) -m ′ N (z) , u N (z) = m RN (z), u ′ N (z) = m ′ RN (z).
Then, M L i=1 ĉi ρi -N i N ρ i can be written as :

M L i=1 ĉi ρi - N i N ρ i = - M 2iN π C z m N (z)X ′ N (z) -u ′ N (z)X N (z) m N (z)u N (z) dz, Υ N (X N , X ′ N , u N , u ′ N ), where Υ N (x, x ′ , u, u ′ ) = - M 2iN π C z m N (z)x ′ (z) -u ′ (z)x(z) m N (z)u(z) dz.
On the other hand, using the decomposition x(z)u(z) -ℓ-1 m N (z) -k+1+ℓ dz.

a k -b k = (a -b) k-1 ℓ=0 a ℓ b k-1-ℓ , we can prove that : L i=1 M ĉi ρk i - N i N ρ k i = M 2 (-1) k 2iN π(k -1) C k-2 ℓ=0 - m RN (z) -m N (z) m ℓ+1 RN (z)m k-1-ℓ N (z) dz = M (-1) k+1 2iN (k -1)π C k-2 ℓ=0 X N (z)u N (z) -ℓ-1 m N (z) -k+1+ℓ dz Φ N,k (X N , u N ),
The main idea of the proof of the theorem lies in the following steps :

1. Prove the convergence of the processes (X N , X ′ N , u N , u ′ N ) and (X N , u N ) over the contour C by using Bai and Silverstein's theorem [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF].

2. Prove the convergence of [Υ N (X N , X ′ N , u N , u ′ N ), Φ N,2 (X N , u N ), • • • , Φ N,L (X N , u N )] T to a Gaussian random vector with the help of the continuous mapping theorem (cf. Theorem 4.2.5).

Compute the limiting covariance between M

L i=1 ĉi ρk i -N i N ρ k i and M L i=1 ĉi ρℓ i -N i N ρ ℓ i .

Conclude by expressing

M ĉ1 -N 1 N , • • • , ĉL -N L N , ρ1 -ρ 1 , • • • , ρL -ρ L T
as a linear func-

tion of M [γ 0 -γ 0 , • • • , γ2L-1 -γ 2L-1 ] T .

Fluctuations of the moments

The next step is to prove the convergence of the vector

Υ N (X N , X ′ N , u N , u ′ N ), Φ N,2 (X N , u N ), • • • , Φ N,L (X N , u N ) T .
The convergence of Υ N (X N , X ′ N , u N , u ′ N ) to a Gaussian random variable has been established in Theorem 4. 1.3 where it has been proved that : The next task is to prove the convergence in distribution of Φ N,k (X N , u N ) over the contour C, for 2 ≤ k ≤ L. Let Φ k (x, u) be defined as :

Φ k (x, u) = (-1) k 2icπ C x(z)u(z) -k dz.
We want to show that Φ k (X N , u N ) converges in distribution to a Gaussian vector. The continuous mapping theorem is useful to transform one convergence to another. Then, since inf C |m| > 0 (see [START_REF] Bai | Spectral Analysis of Large Dimensional Random Matrices[END_REF]Section 9.12]), the dominated convergence theorem implies that the convergence of (x N , y N ) → (x, y) ∈ Γ leads to Φ N,k (x N , y N ) → Φ k (x, y). The continuous mapping theorem applies, thus giving :

Φ N,k (X N , u N ) D ------→ M,N →∞ Φ k (X, u).
It now remains to prove that the limit law Φ k (X, u) is Gaussian. For that, it suffices to notice that the integral can be written as the limit of a finite Riemann sum and that a finite Riemann sum of the elements of a Gaussian random vector is still Gaussian. The convergence of Υ N (X N , X ′ N , u N , u ′ N ) and Φ N,k (X N , u N ) to Gaussian random variables is not sufficient to establish a CLT for the whole vector. It remains to prove that any linear combination of [Υ N (X N , X ′ N , u N , u ′ N ), Φ N,2 (X N , u N ), • • • , Φ N,L (X N , u N )] T converges toward a Gaussian distribution, which can easily be established in the same way as before. It implies that this vector converges to a Gaussian vector. This ends the proof of the fluctuations of the moments.

Computation of the variance

We now come to the third step. We shall therefore evaluate the quantities :

V 1,1 = E Υ(X, Y, m, m ′ )Υ(X, Y, m, m ′ ) , V 1,k = V k,1 = E Υ(X, Y, m, m ′ )Φ k (X, m) , 2 ≤ k ≤ L, V k,ℓ = E [Φ k (X, m)Φ ℓ (X, m)] , 2 ≤ k, ℓ ≤ 2L -1.
The details of the calculations are in Appendix 4.3.5 and yield : For 1 ≤ k, ℓ ≤ 2L -1 Remark 4.2.6 The zeros in the variance simply follow from the fact that γ0γ 0 = 0.

V k,ℓ = - (-1) k+ℓ 4π 2 c 2 C 1 C 2 m ′ (z 1 )m ′ (z 2 ) (m(z 1 ) -m(z 2 )) 2 - 1 (z 1 -z 2 ) 2 1 m k (z 1 )m ℓ (z 2 ) dz 1 dz 2 . ( 4

Fluctuations of the eigenvalues estimates

To transfer this convergence to q M M ĉ1 -

N 1 N , • • • , ĉL -N L N , ρ1 -ρ 1 , • • • , ρL -ρ L T
, we shall use Slutsky's lemma which is as below : Lemma 4.2.2 (cf. [START_REF] Vaart | Asymptotic statistics[END_REF]) Let X n , Y n be sequences of vector or matrix random elements. If X n converges in distribution to a random element X, and Y n converges in probability to a constant C, then

Y -1 n X n D -→ C -1 X
provided that C is invertible.

We will show that w M satisfies the following linear system :

w M = MM q M (4.2.12)
where we will try to find a matrix MM who converges in probability to M which is given by (4.2.5).

To this end, let us work out the expression of w k,M , the k-th element of w M . If k = 1, it is easy to see that w 1,M = 0. For k ≥ 2, w k,M is given by :

w k,M = M L i=1 ĉi ρk-1 i - N i N ρ k-1 i = M L i=1 ĉi ρk-1 i - N i N ρk-1 i + N i N ρk-1 i - N i N ρ k-1 i = M L i=1 ĉi - N i N ρk-1 i + N i N (ρ i -ρ i ) k-2 ℓ=0 ρℓ i ρ k-2-ℓ i .
Then define

MM =       1 • • • 1 0 • • • 0 ρ1 • • • ρL N 1 N • • • N L N . . . . . . . . . . . . . . . . . . ρ2L-1 1 • • • ρ2L-1 L N 1 N 2L-2 ℓ=0 ρℓ 1 ρ 2L-2-ℓ 1 • • • N L N 2L-2 ℓ=0 ρℓ L ρ 2L-2-ℓ L       .
We can easily check that Eq. (4.2.12) is satisfied and MM converges in probability to M. It remains to check that M is invertible. Note that the non-singularity of matrix M has already been established in Section 4.2.2, where this property was required to prove the existence of an estimator. As a consequence, using Slutsky's lemma, we deduce that : This ends the proof for the fluctuation.

Simulations

In this section, we compare the performance of the proposed estimator with Mestre's estimator [START_REF] Mestre | Improved estimation of eigenvalues of covariance matrices and their associated subspaces using their sample estimates[END_REF] in Section 4.2.4 ; we then compare the proposed estimator with the estimator proposed by Bai et al. [START_REF] Bai | On estimation of the population spectral distribution from a high-dimensional sample covariance matrix[END_REF] in Section 4.2.4. We finally verify by simulations the accuracy of the Gaussian approximation stated by the CLT in Section 4.2.4.

Comparison with the estimator in [START_REF] Mestre | Improved estimation of eigenvalues of covariance matrices and their associated subspaces using their sample estimates[END_REF] As will be seen below, the separability assumption is compulsory for Mestre's method to be effective. If this assumption holds true, a simple clustering procedure enables to estimate the unknown multiplicities and Mestre's method outperforms our moment estimator (see Fig. 4.4).

If, however, the separability assumption is not met, then it is not clear how to directly estimate (even roughly) the multiplicities ; and even if those were known, Mestre's estimation method has no methodological foundations (as the estimator is not even consistent in this case !) and the computation of Mestre's estimator yields a systematic error (see Fig. 4.5 for instance).

A final remark is in order with respect to the separability assumption : Although it is easy in simulations to generate data fulfilling or violating the separability assumption, it is not an easy task, while facing real data, to decide whether the separability assumption holds true or not. Building such a test remains an open problem, advocating for our procedure by defaultunless any extra argument emerges to support a separability assumption. Otherwise stated, the non-separability assumption is much more realistic in practical cases.

In the first experiment, we consider the case where the separability condition holds true. We assume also that the covariance matrix has three different eigenvalues (ρ 1 , ρ 2 , ρ 3 ) = (1, 3, 7), which are distributed as :

N 1 N = 0.5, N 2 N = N 3 N = 1 4 .
The ratio N M is set to 30 200 = 3 20 . The separability condition being met, the clusters are well separated so that the multiplicities can be estimated in a heuristic way based on the difference of the ordered eigenvalues. More precisely, an empirical method for estimating the multiplicities consists in the following steps :

-Arrange the eigenvalues of the covariance matrix in increasing order : λ1 ≤ • • • , ≤ λN .

-Take L indexes i 1 , • • • , i L satisfying : -Arrange these indexes in the increasing order : i

[1] ≤ • • • ≤ i [L]
. Empirical estimates of the multiplicities are thus given by :

N1 = i [1] N2 = i [2] -i [1]
. . .

NL = N -i [L-1] .
This empirical method has proved to be efficient in the asymptotic regime. For example, by applying exact separation results from [START_REF] Bai | No Eigenvalues Outside the Support of the Limiting Spectral Distribution of Large Dimensional Sample Covariance Matrices[END_REF][START_REF] Bai | Exact Separation of Eigenvalues of Large Dimensional Sample Covariance Matrices[END_REF], it can be proved that the estimates of the normalized multiplicities ( Nk /N ) are asymptotically consistent. Fig. 4.4 compares the performance of the Mestre's estimator using the aforementioned method for estimating the multiplicities with that of the proposed estimator, in terms of MSE : In this case, Mestre's estimator outperforms the proposed estimator. This can be attributed to numerical difficulties which will be discussed in the next section.

In the second experiment, we consider the case where the separability condition does not hold. In particular, we assume that the covariance matrix R N has three different eigenvalues (ρ 1 , ρ 2 , ρ 3 ) = (1, 2, 3), each with the same multiplicity, i.e. N 1 N = N 2 N = N 3 N = 1 3 . We also set the ratio between the dimension of variables and the number of samples N M to 3/8, a ratio which is too high for the separability condition to hold true. We assume for our estimator that the multiplicities are not known, a hypothesis that obviously cannot be used for Mestre's estimator. We thus favour Mestre's estimator by assuming that it knows perfectly the multiplicities. Fig. 4.5 compares the obtained results in terms of MSE : for different values of M and N satisfying a constant ratio c = N/M = 3/8 and 1000 realizations. We note that as M and N increase, the estimator in [START_REF] Mestre | Improved estimation of eigenvalues of covariance matrices and their associated subspaces using their sample estimates[END_REF] exhibits an error floor, underlying the fact that without the separability assumption, Mestre's estimators are no longer consistent.

Comparison with the method in [START_REF] Bai | On estimation of the population spectral distribution from a high-dimensional sample covariance matrix[END_REF] The estimator proposed in [START_REF] Bai | On estimation of the population spectral distribution from a high-dimensional sample covariance matrix[END_REF] and our proposed estimator are similar at first sight. The main difference lies in the intermediate quantities which are estimated before estimating the eigenvalues and their multiplicities. While the technique of [START_REF] Bai | On estimation of the population spectral distribution from a high-dimensional sample covariance matrix[END_REF] is based on the numerical computation of the empirical moments 1 N Tr(Y N Y H N ) k , our technique rather relies on building consistent estimators of the theoretical moments 1 N TrR k N . This difference induces important numerical consequences in the computation of the estimates : In [START_REF] Bai | On estimation of the population spectral distribution from a high-dimensional sample covariance matrix[END_REF], the functional relation between the quantities to be estimated and the empirical moments 1 N Tr(Y N Y H N ) k yields a system of equations whose resolution relies on iterative methods (based for instance on the functions fsolve or fminsearch in Matlab) which are extremely slow.

On the other hand, the method proposed in this article is based on a bijective system of equations that links the theoretical moments to the eigenvalues and their multiplicities, whose resolution relies on simple computations : A matrix inversion and solving a polynomial (see for instance end of Section 4.2.2).

Simulation results indicate that our algorithm allows a great gain of complexity compared to [START_REF] Bai | On estimation of the population spectral distribution from a high-dimensional sample covariance matrix[END_REF], while keeping the same level of performance. Execution times for one realization are provided in the following table 4.1 for the same simulation setting as the second experiment. Note that unlike our method which exhibits low complexity, the complexity of the method of [START_REF] Bai | On estimation of the population spectral distribution from a high-dimensional sample covariance matrix[END_REF]

tends to

The following lemma clarifies the relations between S N and S. Lemma 4.3.1 Under Assumptions A1 and A2, for N large enough, the support S N of the probability distribution associated to the Stieltjes transform m N (z) is the union of L clusters :

For a N 1 ≤ b N 1 < • • • < a N L ≤ b N L , S N = [a N 1 , b N 1 ] ∪ • • • ∪ [a N L , b N L ] .
Moreover, the following convergence holds true : N → c 1 > 0, hence zero is also in the support S. In this case, we will get that a 1 = b 1 = 0, and the conclusion still holds true.

a N ℓ ------→ N,M →∞
Proof :[Proof of Lemma 4.3.1] Recall the relations : By multiplying the common denominator, one gets a polynomial of the degree 2L in m a . Let us now prove that these 2L roots are real. At first, note that :

m N (z) = -z - N M t 1 + tm N F R N (dt)
1 m 2 - N M t 2 dF R N (t) (1 + tm) 2 -----→ m→-1 ρ i -∞, and
z ′′ R N (m) = - 2 m 3 + N M 2t 3 dF R N (t) (1 + tm) 3 .
So z ′′ R N (m) has one and only one zero in the open set (-

1 ρ i , -1 ρ i+1 ) for 1 ≤ i ≤ L -1 . Then for β i ∈ (-1 ρ i , -1 ρ i+1
) such that z ′′ R N (β i ) = 0, it suffices to show that z ′ R N (β i ) > 0 in order to prove that there will be two zeros for z ′ R N (m) in the set (-1 ρ i , -1 ρ i+1 ). From the separability condition (cf. Assumption A2), inf N { M N -Ψ N (i)} > 0, and The second term of the right hand side (r.h.s.) of the equation can be handled by the induction hypothesis :

z ′ R N - 1 α i = α 2 i - N M t 2 dF R N (t) (1 -t α i ) 2 , = α 2 i 1 - 1 M L r=1 N i ρ 2 i (α i -ρ i ) 2 > 0 .
E[Trf ( RN )] 2 k 2 = O 1 N 2β .
We now rely on Poincaré-Nash inequality (see for instance [41, Section II-B]) to handle the first term of the r.h.s. Applying this inequality, we obtain :

Var (Trf ( RN )) 2 k ≤ K i,j E   ∂[Trf ( RN )] 2 k ∂Y i,j 2 + ∂[Trf ( RN )] 2 k ∂Y i,j 2   , (4.3.5)
where K is a constant which does not depend on N, M and which is greater than R N 's eigenvalues. In order to compute the derivatives of the r.h.s., we rely on [START_REF] Haagerup | A new application of random matrices : Ext(C * red (F 2 )) is not a group[END_REF]Lemma 4.6]. This yields :

∂ ∂Y i,j [Trf ( RN )] 2 k = 2 k M [Trf ( RN )] 2 k -1 [Y * N f ′ ( RN )] j,i , ∂ ∂Y i,j [Trf ( RN )] 2 k = 2 k M [Trf ( RN )] 2 k -1 [f ′ ( RN )Y N ] i,j .
Plugging these derivatives into (4.3.5), we obtain :

Var(Tr[f ( RN )] 2 k ) ≤ K 2 2k+1 M 2 E (Trf ( RN )) (2 k+1 -2) Tr(f ′ ( RN )Y N Y * N f ′ ( RN )) , = K 2 2k+1 M E (Trf ( RN )) (2 k+1 -2) Tr(f ′ ( RN ) 2 RN ) , ≤ K 2 2k+1 M E[Trf ( RN )] 2 k+1 2 k+1 -2 2 k+1 × E[Trf ′ ( RN ) 2 RN ] 2 k 1 2 k ,
where the last inequality is a consequence of Hölder's inequality.

As the function h(λ) = λ[f ′ (λ)] 2 satisfies the induction hypothesis, we have for every α < 1 :

ETr[f ′ ( RN ) 2 RN ] 2 k 1 2 k = O(N -α ).
Plugging this estimate into (4.3.4), we obtain :

E[Trf ( RN )] 2 (k+1) ≤ K 1 N 1+α |E[Trf ( RN )] 2 (k+1) | 2 k+1 -2 2 k+1
+ O(N -2β ) ,

E[Trf ( RN )] 2 (k+1) = O N a ℓ ∨(-2β) ,
where a ℓ = a ℓ-1 2 k+1 -2 2 k+1 -(1 + α) and x ∨ y stands for sup(x, y). Now, in order to conclude the proof, it remains to prove that i) the sequence (a ℓ ) converges to some limit a ∞ , ii) for some well-chosen α < 1, a ∞ ∈ (-2 k+1 , -2β). Write : Notice that X N (z) = M (m RN m N ) = X N (z) for z ∈ C + . So it suffices to verify the arguments for z ∈ C + . As 1 ρk -z can converge to infinity if z is close to the real axis, the process X N (z) might be large when z is close to the real axis. Thus we begin the proof by considering a truncated version of the process X N . More precisely, let ε N be a real sequence decreasing to zero satisfying for some δ ∈]0, 1[ :

a ℓ+1 + 2 k (1 + α) = 2 k -1 2 k (a ℓ + 2 k (1 + α)) ,
ε N ≥ N -δ .
With the same notations as in Lemma 4.

z 1 -z 2 = 1 M [-D -2 N (z 1 )D -1 N (z 2 )Y N -D -1 N (z 1 )D -2 N (z 2 )Y N ] i,j .
Then by the Poincaré-Nash inequality and the fact that RN is uniformly bounded in spectral norm almost surely, one gets The second condition to establish the tightness is achieved. For X2 N (z), following exactly the same method as in [START_REF] Bai | Spectral Analysis of Large Dimensional Random Matrices[END_REF]Section 9.11], one can prove that X2 N (z) is bounded and forms an equicontinuous family that converges to 0. Hence the tightness for M (m RN (z)m N (z)). .

E| X1 N (z 1 ) -X1 N (z 2 )| 2 |z 1 -z 2 | 2 ≤ C 1 N E Tr(L N,1 ) = C 1 N E(
We thus obtain : .3.15) This proves that the theoretical moments admit the following integral representation :

L k=1 N k N ρ ℓ k = M (-1) ℓ 2iπN (ℓ -1) C dz m ℓ-1 N (z) . ( 4
γ 1 = - M 2N iπ C zm ′ N (z) m N (z) dz γ l = M (-1) ℓ 2iN (ℓ -1) C dz m l-1 N (z) , 2 ≤ ℓ ≤ 2L -1.
Finally, we show that consistent estimates of γ i can be obtained by substituting the unknown term m N (z) by its asymptotic equivalent m RN (z). Let γ0 , • • • , γ2L-1 the real quantities given by : γ0 = 1, --→ 0.

Consequently :

γiγ i a.s.

------→ N,M →∞ 0.

Calculation of the variance

In this section, we will show the calculations of the variance matrix V. The computation of V 1,1 has been carried out in Theorem 4.1.3 where it was shown that :

V 1,1 = - 1 4π 2 c 2 C 1 C 2 m ′ (z 1 )m ′ (z 2 ) (m(z 1 ) -m(z 2 )) 2 - 1 (z 1 -z 2 ) 2 1 m(z 1 )m(z 2 ) dz 1 dz 2 ,
with C 1 and C 2 defined in the theorem. Using the fact that inf z∈C |m(z)| > 0 together with Fubini's theorem, the quantity V k,ℓ for k ≥ 2, ℓ ≥ 2, becomes :

V k,ℓ = - (-1) k+ℓ 4π 2 c 2 C 1 C 2 E [X(z 1 )X(z 2 )] m -k (z 1 )m -ℓ (z 2 )dz 1 dz 2 .
Substituting E [X(z 1 )X(z 2 )] by κ(z 1 , z 2 ), we obtain :

V k,ℓ = - (-1) k+ℓ 4π 2 c 2 C 1 C 2 m ′ (z 1 )m ′ (z 2 ) (m(z 1 ) -m(z 2 )) 2 - 1 (z 1 -z 2 ) 2 1 m k (z 1 )m ℓ (z 2 ) dz 1 dz 2 .
Finally, it remains to compute V k,1 . Expanding Υ(X, Y, m, m ′ ) and Φ k (X, m), we obtain :

V k,1 = - (-1) k+1 4π 2 c 2 C 1 C 2 z 2 m(z 2 )m k (z 1 ) E X(z 1 )X ′ (z 2 ) dz 1 dz 2 - m ′ (z 2 ) m(z 2 ) 2 m k (z 1 ) E [X(z 1 )X(z 2 )] dz 1 dz 2 = - (-1) k+1 4π 2 c 2 C 1 C 2 z 2 ∂ 2 κ(z 1 , z 2 ) m(z 2 )m(z 1 ) k dz 1 dz 2 - C 1 C 2 m ′ (z 2 )κ(z 1 , z 2 ) m 2 (z 2 )m k (z 1 ) dz 1 dz 2 .
By integration by parts, we obtain :

C 2 z 2 ∂ 2 κ(z 1 , z 2 ) m(z 2 )m k (z 1 ) dz 2 = - C 2 κ(z 1 , z 2 ) m(z 2 )m k (z 1 ) dz 2 + C 2 m ′ (z 2 )κ(z 1 , z 2 ) m(z 2 ) 2 m k (z 1 ) dz 2 .
Hence,

V k,1 = - (-1) k+1 4π 2 c 2 C 1 C 2 κ(z 1 , z 2 )dz 1 dz 2 m(z 2 )m k (z 1 ) .
This extends the expression of V k,ℓ for any k, ℓ ∈ {1, • • • , L -1}, thus yielding :

V k,ℓ = - (-1) k+ℓ 4π 2 c 2 C 1 C 2 m ′ (z 1 )m ′ (z 2 ) (m(z 1 ) -m(z 2 )) 2 - 1 (z 1 -z 2 ) 2 1 m k (z 1 )m ℓ (z 2 )
dz 1 dz 2 . (4. 3.16)

  and distribution CN (m, Q) complex Gaussian distribution with mean m and covariance Q N (m, Q) Gaussian distribution with mean m and covariance Q X(ω) the realization of the variable X at point ω δ x the Dirac probability measure at x Analysis C, R, N the complex, real and natural number Re(z), Im(z) real and imaginary part of z z conjugate of z C + {z ∈ C : Im(z) > 0} R + , R - nonnegative and nonpositive real numbers

- 1

 1 .1.3) et T n (resp. Tn ) l'équivalent déterministe de Q n (resp. Qn ) défini par    T n (z) = -z(1 + δn )I N + AnA * n 1+δn Tn (z) = -z(1 + δ n )I n + A * n An 1+ δn -1

1 N

 1 où X n (M ) est une matrice N M × nM centrée réduite et A n (M ) est une matrice par bloc dont les éléments sur la diagonale par bloc est A n . L'intérêt d'introduire la matrice A n (M )repose sur le fait que Y n (M )Y n (M ) * a le même équivalent déterministe que Y n Y * n , i.e., 1 N M TrQ nM (z) -TrT n (z) ----→ M →∞ 0 où Q nM = (Y n (M )Y n * (M ) -zI N M ) -1 . δ nM =δ n est indépendant de M du fait de la nature bloc-diagonale de A n (M ). N, n étant fixés, notons N (B n,M , Θ n,M ) la famille gaussienne de la loi limite associée à Q nM , nous pouvons montrer sans difficulté que ∀M ≥ 1, B n,M = B n , Θ n,M = Θ n , et transférer les propriétés de TrQ n aux B n et Θ n . Cette méthode sera utilisée de manière systématique dans l'étude de CLT (cf. Section 2.3.6, 3.3.1).

1 N 1 M 1 N 1 N

 1111 Tr(R N + xI N ) -1 où x ≥ 0. Ce problème est important car de nombreux autres estimateurs sont basés sur celui de 1 N Tr(R N + xI N ) -1 . En absence d'hypothèse sur la loi de probabilité des vecteurs observés, l'usage consiste à estimer R N par la matrice de covariance empirique RN définie par Tr(R N +xI N ) -1 par 1 N Tr( RN +xI N ) -1 . Cette pratique est essentiellement justifiée par le fait que si n → ∞ et que N reste fixe, cet estimateur est consistant, c'est-à-dire que lim n→∞ Tr( RN + xI N ) -1 = Tr(R N + xI N ) -1 .

Lemma 2 . 4 . 4

 244 Recall the definition of n j=1 ξ 2j and n j=1 ξ ′ 2j in (2.4.26) and (2.4.27) respectively, we have

  ℓ a k are uncorrelated and (b) is from Lemma 2.4.3 and Equation (2.4.15) (cf. Remark 2.4.5).

Theorem 3 . 1 . 1 8 )

 3118 Under Assumptions 3.1.1, 3.1.2 3.1.3, let k ∈ N and f be an integrable function over R satisfying R | f (t)|(1 + |t| 4 )dt < ∞. (3.1.Recall the definition of tn in (3.1.2) with the support S n and the random vector N 1 n (f ) in(3.1.4)

.

  The expression of the covariance coincides with[START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF] Eq. (1.20)] and [8, Eq. (1.5)]. Remark 3.1.1 Loosely speaking, condition (3.1.8) implies that de derivatives f (k) is bounded over R for k = 0, 1, 2, 3, 4.

.1. 11 ) 3 . 1 . 2

 11312 Theorem Let f be a function of class C 18 , integrable on R satisfying

Proposition 3 . 3 . 1 [ 70 ]

 33170 Let k ∈ N and f be a function which is analytic on an open region containing [lim inf n λ Rn minI 0≤c≤1 (1 -√ c) 2 , lim sup n λ Rn max (1 + √ c) 2 ]. (3.3.3) N (m n (z) -Em n (z)). By Cauchy formula, ůn (x) = N k=1 e iλ k x -Ee iλ k x = 1 2iπ Cn e izx M 1 n (z)dzwhere the contour C n is a rectangle in trigonometric direction which contains(3.3.3). As g(z) = e izx is clearly analytic over C, by Proposition 3.3.1,

By a simple calculation

  of the bloc matrix, Y n (M )Y n * (M ) has the same deterministic equivalent t n as Y n Y n * . With Proposition 3.3.1, we can show that for n, N fixed, M 1 n,M (z) converges to a Gaussian process with

R

  n < ∞ by Assumption 3.1.3. Then (3.3.5) holds true for u M n (t) = N M j=1 e itλ j where λ j are eigenvalues of Y n (M )Y T n (M ) :

.4. 30 )

 30 Subtracting (3.4.29) and (3.4.30), we have

F

  (z)dz = 0, where [α, β] denotes the line segment connecting α, β in C oriented from α to β. Then as | Mc (z + w)| ≤ 1 |Imw| for z ∈ C + , we get [r,r+ir]

3. 5 . 1

 51 Proof of Lemma 3.1.1

3. 5 . 5 4 ℓ = 1 , 2

 55412 Proof of Proposition 3.2.is trivial. We shall prove the proposition for ℓ = 3 or 4. When ℓ = 3,

Remark 3 . 5 . 1

 351 Readers should pay attention that the polynomials C ℓ here are not the same. It is clear that for z ∈ E, (3.4.25) holds. To show (3.4.26), we have |det(I -D 0 )| = |1z 2 v 0 ṽ0 | = 1 -|z| 2 |v 0 ||ṽ 0 |. By Cauchy-Schwarz, |v 0 | ≤ |v| 1/2 |v ′ | 1/2 and |ṽ 0 | ≤ |ṽ| 1/2 |ṽ ′ | 1/2 , which yield that

Figure 4 .

 4 1 depicts the eigenvalues of a realization of the random matrix RN and the associated limiting distribution as N, M grow large, for ρ 1 = 1, ρ 2 = 3, ρ 3 = 10 and N = 60, M = 600 with N 1 = N 2 = N 3 = 20. The separability condition is illustrated there.

  Figure 4.2 shows another situation where the separability condition is not satisfied for ρ 1 = 1, ρ 2 = 3, ρ 3 = 5 and N = 30, M = 80 with N 1 = N 2 = N 3 = 10.

Theorem 4 . 1 . 1 (-

 411 [START_REF] Mestre | Improved estimation of eigenvalues of covariance matrices and their associated subspaces using their sample estimates[END_REF] Th. 3]) Let Assumptions A1 and A2 hold true and denote by λ1 ≤ • • • ≤ λN the ordered eigenvalues of RN . Then the following convergence holds true :ρkk λmμm ,(4.1.4)withN k = { k-1 j=1 N j + 1, . . . , k j=1 N j }and the μi 's defined 1 as follows : -If N ≤ M , then μ1 ≤ • • • ≤ μN are the real ordered solutions of If N > M , μi = 0 for 1 ≤ i ≤ N -M and μN-M+1 , • • • , μN are the real solutions of the above equation.

1 .

 1 Another characterization of interest of the μi's is the fact that they are the eigenvalues of diag( λ) -1 M λ λT, where λ = ( λ1, . . . , λN ) T , see for instance[START_REF] Couillet | Random matrix methods for wireless communications[END_REF] Chapter 8].

Figure 4 . 1 -

 41 Figure 4.1 -Empirical and asymptotic eigenvalue distribution of RN for L = 3,ρ 1 = 1, ρ 2 = 3, ρ 3 = 10, N/M = c = 0.1, N = 60, N 1 = N 2 = N 3 = 20.

Figure 4 . 2 -

 42 Figure 4.2 -Empirical and asymptotic eigenvalue distribution of RN for L = 3,ρ 1 = 1, ρ 2 = 3, ρ 3 = 5, N/M = c = 3/8, N = 30, N 1 = N 2 = N 3 = 10.

Theorem 4 . 1 . 2 (

 412 [START_REF] Silverstein | On the empirical distribution of eigenvalues of a class of large dimensional random matrices[END_REF]) Let Assumption A1 hold true and denote by F R the limiting spectral distribution of R N , i.e.F R (d λ) = L k=1 c k c δ ρ k (d λ) .

Remark 4 . 1 . 1

 411 Denote by m RN (z) and m RN (z) the Stieltjes transforms of F RN and F RN . Note in particular that m RN (z) = This relation associated to (4.1.5) readily implies that for μi = 0, m RN (μ i ) = 0. Otherwise stated, the (non null) μi 's are the zeros of m RN

1 . 7 )

 17 satisfying m N (z) ∈ C + if z ∈ C + . It can be shown that m N and m N are Stieltjes transforms of probability measures F N and F N , respectively (cf. [25, Theorem 3.2]).

Theorem 4 . 1 . 3

 413 Let Assumptions A1 and A2 hold true and recall the definitions of the m(z) and m N (z) in Section 4.1.6 and 4.1.7 respectively. Then :

Remark 4 . 1 . 3

 413 In Theorem 4.1.3, the separability assumption A2 can be relaxed to some extent. For example, if only the cluster associated to ρ k satisfies the separability condition, one can study the fluctuations of ρk by relying on the same techniques.Proof of Theorem 4.1.3

Remark 4 . 1 . 7

 417 Due to the properties of the process (X, Y ), E X(z)X(w) = κ(z, w) (and similarly for the other cross-conjugate quantities E X(z)Y (w), etc.) ; moreover, the quantities E X(z)Y (z) and E Y (z)Y (z) can be computed by considering the limits lim z→z ∂κ ∂z (z, z) and lim z→z ∂ 2 κ ∂z∂ z (z, z). The covariance structure of the process (X, Y ) is hence fully described.

Lemma 4 .

 4 1.1 is the cornerstone to the proof of Theorem 4.2.1 ; its proof is postponed to Appendix 4.3.2 and relies on the following proposition, of independent interest : Proposition 4.1.1 Assume that A1 and A2 hold true and denote by S the support of the probability distribution associated to the Stieltjes transform m. Then, for every ε > 0, ℓ ∈ N * : P sup λ∈eig( RN ) d(λ, S) > ε = O 1 N ℓ , where d(λ, S) = inf x∈S |λ -x|. The proof of Proposition 4.1.1 is postponed to Appendix 4.3.1. As (u N , u ′ N ) a.s.

Theorem 4 .

 4 1.3 describes the limiting performance of the estimator of Theorem 4.1.1, with an exact characterization of its variance. Unfortunately, the variance Θ depends upon unknown quantities. We provide hereafter consistent estimates Θ for Θ based on the observation RN .

Theorem 4 . 1 . 5

 415 Assume that Assumptions A1 and A2 hold true, and recall the definition of Θ kℓ given in (4.1.10) and Theorem 4.2.1. Let Θkℓ be defined by

  where (N k ) and (μ k ) are defined in Theorem 4.1.1, then :Θkℓ -Θ kℓ a.s.--→ 0 as N, M → ∞.

Theorem 4 .

 4 1.5 is useful in practice as one can obtain simultaneously an estimate ρk of the values of ρ k as well as an estimation of the degree of confidence for each ρk . Proof : In view of formula (4.1.10), taking into account the fact that m RN and m ′ RN are consistent estimates for m and m ′ , it is natural to define Θkℓ by replacing the unknown quantities m and m ′ in (4.1.10) by their empirical counterparts m RN and m ′ RN , hence we define Θkℓ as Θkℓ

( 4 . 1 . 15 )

 4115 The proof of Theorem 4.1.5 now breaks down into two steps : The convergence of Θkℓ to Θ kℓ , which relies on the definition (4.1.15) of Θkℓ and on a dominated convergence argument, and the effective computation of the integral in (4.1.15) which relies on Cauchy's residue theorem[START_REF] Marsden | Basic Complex Analysis[END_REF], and yields (4.1.14).

Proposition 4 . 1 . 2

 412 Let Assumptions A1 and A2 hold true, then for z 2 ∈ R ′ ℓ \R, almost surely,

Figure 4 . 3 -

 43 Figure 4.3 -Comparison of empirical against theoretical variances, based on Theorem 4.2.1, for three eigenvalues, ρ 1 = 1, ρ 2 = 3, ρ 3 = 10, N 1 = N 2 = N 3 = 20, N = 60, M = 600 and SNR= 20 dB.

Assumption 4 . 2 . 4

 424 The support S of the limiting probability distribution of F RN is composed of L compact connex disjoint subsets, and not reduced to a singleton. (cf. Section ?? and Fig.4.1 and 4.2) Remark 4.2.2 Note that when M < N , matrix RN is singular and thus admits (N -M ) eigenvalues equal to zero. Hence, the limiting spectrum of RN has an additional mass in zero with weight 1 -1 c , which will not be considered among the L clusters.Recall the notations in Theorem 4.1.2 with the Stieljes transform m(z) (resp. m(z)) of the LSD of RN (resp. RN = 1 M Y H N Y N )and the deterministic equivalent m N (z) (resp. m N (z)). In[START_REF] Mestre | Improved estimation of eigenvalues of covariance matrices and their associated subspaces using their sample estimates[END_REF], Mestre proposes a novel approach to estimate the eigenvalues (ρ k ; 1 ≤ k ≤ L) of the population covariance matrix based on the observations RN under the additional Assumption 4.2.4. His approach relies on large random matrix theory and the separability condition presented above plays a major role in the mere definition of the estimators. Recall the estimator (cf. Theorem 4.1.1 ) ρkρ k a.s. k λmμm . A CLT has been derived in Theorem 4.1.3 for this estimator under the extra assumption that the entries of X N are Gaussian. To unify the notation, recall again the theorem : Theorem 4.2.1 ([99]) With the same notations as before, under Assumptions 4.2.1, 4.2.2, 3b, 4.2.4 and with known multiplicities N 1 , • • • , N L , then :

3 .

 3 Compute the vector s as s = -Γ -1 b.

Theorem 4 . 2 . 4

 424 Let Assumptions 4.2.1, 4.2.2, 3b hold true. Let (ĉ 1 , • • • , ĉL , ρ1 , • • • , ρL ) be the estimators obtained in Theorem 4.2.2. Then

for 2 ≤

 2 k ≤ 2L -1, where Φ N,k (x, u) = M (-1) k+1 2iN (k -1)π C k-2 ℓ=0

Υ

  N (X N , X ′ N , u N , u ′ N ) D ------→ M,N →∞ Υ(X, Y, m, m ′ ) where Υ(x, y, v, w) = 1 2iπc C z m(z)y(z)w(z)x(z) m(z)v(z) dz.

Theorem 4 . 2 . 5 (

 425 cf.[START_REF] Kallenberg | Foundations of mordern Probability[END_REF] Th. 4.27]) For any metric spaces S 1 and S 2 , let ξ, (ξ n ) n≥1 be random elements in S 1 with ξ n D ---→ n→∞ ξ and consider some measurable mappings f , (f n ) n≥1 : S 1 → S 2 and a measurable set Γ ⊂ S 1 with ξ ∈ Γ a.s. such thatf n (s n ) → f (s) as s n → s ∈ Γ. Then f n (ξ n ) D ---→ n→∞ f (ξ).Consider the set : Γ = (x, u) ∈ C 2 (C, C) , inf C |u| > 0 .

.2. 11 )

 11 Let w M = M [γ 0γ 0 , • • • , γ2L-1γ 2L-1 ] T .We have just proved that the vector w M converges asymptotically to : V is the (2L -1) × (2L -1) matrix whose entries V k,l are given by (4.2.11).

N

  2L 0, M -1 W(M -1 ) T .

i 1 Figure 4 . 4 -

 144 Figure 4.4 -Experienced MSE with N when N M = 3 20 and (ρ 1 , ρ 2 , ρ 3 ) = (1, 3, 7)

Figure 4 . 5 -

 45 Figure 4.5 -Experienced MSE with N when N M = 3 8 and (ρ 1 , ρ 2 , ρ 3 ) = (1, 2, 3)

for 1 ≤

 1 ℓ ≤ L. Remark 4.3.1 If the support S N contains zero, (ex : N > M ), a N 1 = b N 1 = 0. By Assumption A1, the multiplicity N 1 corresponding to zero satisfies N 1

2 ) 2 2 ρ 2 i( 1 +

 22221 As the Stieltjes transform of δ 0 (the Dirac mass at 0) is -1 z and m N (z) is a continuous function over R * + , for a,b with 0 < a < b, by the inverse formula of Stieltjes transform, one gets :F N ([a, b]) = M N F N ([a, b]).So it suffices to study the support S N associated to F N .From the definition of m N (z) (see formula (4.3.1)), we obtain :z R N (m N ) = -Denote by B = {m ∈ R : m = 0, -m -1 / ∈ {ρ 1 , • • • , ρ L }}.From Lemma[START_REF] Silverstein | Analysis of the limiting spectral distribution of large dimensional random matrices[END_REF], for a real numberx, x ∈ S c N ⇐⇒ m x ∈ B and z ′ R N (m x ) = 1 dF R N (t) (1+tm x ) 2 > 0 with m N (x) = m x and z R N (m x ) = x. Then if a ∈ ∂S N , m a / ∈ B or z ′ R N (m a ) ≤ 0 with m a = m N (a). Now we will show that m a ∈ B. With the same argument in[START_REF] Silverstein | Analysis of the limiting spectral distribution of large dimensional random matrices[END_REF],m a = 0. If -m -1 a ∈ S F R N , as F R N is discrete, we get that lim m→ma t 2 dF R N (t) (1+tm) 2 -→ ∞.So on the neighborhood to the left and to the right of m a , z ′ R N < 0 which contradicts [84, Lemma 3.5]. Hence z ′ R N (m a ) ≤ 0. By the continuity, we get z ′ R N (m a ) dF R N (t) (1 + tm a ) 2 = 0. This is equivalent to the following equation : ρ i m a ) 2 = 0. (4.3.3)

Thus we obtain 2 (L - 1 ) 2 ≤ 2

 2122 roots. Besides, in the open set (-ρ -1 L , root in this set. In the open set (-∞, -ρ Hence the last root in this open set. This proves that SN = [a N 1 , b N 1 ] ∪ • • • ∪ [a N L , b N L ]. To prove a N ℓ ------→ N,M →∞ a ℓ and b N ℓ ------→ N,M →∞ b ℓ , note that a i b i satisfy the same type of equation = E [Trf ( RN )] 2 k + E[Trf ( RN )] 2 k -E[Trf ( RN )] 2 k Var[Tr f ( RN )] 2 k + |E[Trf ( RN )] 2 k | 2 . (4.3.4)

( 4 .2 k+1 - 2 2

 42 3.6) where K is a constant independent of M, N, k. Notice that inequality (4.3.6) involves twice the quantity of interest E[Trf ( RN )] 2 (k+1) that we want to upper bound by O(N -2β ). We shall proceed iteratively.Notice that Tr[f ( RN )] ≤ sup x∈R |f (x)| × N because f is bounded on R ; hence the rough estimate :E[Trf ( RN )] 2 (k+1) = O(N 2 k+1 ).Plugging this into (4.3.6) yields :E[Trf ( RN )] 2 (k+1) = O(N a 1 ) ,where a 0 = 2 k+1 and a 1 = a 0 k+1 -(1 + α). Iterating the procedure, we obtain :

hence a ℓ converges to - 2 k

 2 (1 + α) which readily belongs to (-2 k+1 , -2β) for a well-chosen α ∈ (0, 1). Finally E[Trf ( RN )] 2 (k+1) = O(N -2β ) which ends the induction.It remains to apply this estimate to E[Trφ( RN )] ℓ in order to get the desired result.

4. 3 . 2

 32 Proof of Lemma 4.1.1

3 . 1 , 2 ) z 1 -z 2 = 1 = 1 M -Y * N D - 2 N (z 1 )D - 1 N (z 2 )

 31212112112 denote by S = [a 1 , b 1 ] ∪ • • • ∪ [a L , b L ] ; and take p k , q k such that b k-1 < p k < a k and b k < q k < a k+1 for 1 ≤ k ≤ L with conventions b 0 = 0 and a L+1 = ∞, i .e. [p k , q k ] only contains the k-th cluster. Let d > 0. Consider : R k,1 = {x + id : x ∈ [p k , q k ]} , R k,2 = p k + iv : v ∈ ε N N , d , R k,3 = q k + iv : v ∈ ε N N , d , with D N (z) = RN -zI M . We have ∂ ∂Y i,j m RN (z 1 )m RN (z ∂ ∂Y i,jTr( RN -z 1 I M ) -1 ( RNz 2 I M ) --Y * N D -1 N (z 1 )D -2 N (z 2 ) j,i ,and∂ ∂ Ȳi,j m RN (z 1 )m RN (z 2 )

.N 7 ǫ 6 N

 6 Tr(L N,1 )I sup n d( λ′ n ,S)≤ε ) +C 1 N E(Tr(L N,1 )I sup n d( λ′ n ,S)>ε ) with L N,1 = RN D -4 N (z 1 )D -2 N (z 2 ) + 2 RN D -3 N (z 1 )D -3 N (z 2 ) + RN D -2 N (z 1 )D -4N (z 2 ) and C 1 a constant which does not depend on N or M . For the first term, Tr(L N,1 ) is bounded on the set sup n d( λ′ n , S) ≤ ε. For the second term, since for all i ∈ N and all z ∈ Rk ,1 | λ′ n -z| i ≤ N i As P(sup d( λ′ n , S) ≥ ε) = P(sup d( λn , S) ≥ ε) = o(N -16 ), take ε N = N -0.01 , one obtains E(Tr(L N,1 )I sup n d( λ′ n ,S)>ε ) ≤ E Tr(L N,1 )I sup d( λ′ n ,S)>ε ≤ O P(sup d( λ′ n , S) > ε) → 0.

  The second term on the right hand side of (4.3.14) is then equal to zero. It remains thus to deal with C zm ′ N (z) m ℓ N (z) . If ℓ ≥ 2, by integration by parts, we obtain :

.

  Then, by the dominated convergence theorem and the fact that with probability one [5, Section 9.12], for all N, M large enough, infz∈C |m N (z)| > 0 and inf z∈C |m RN (z)| > 0,one obtains : for all k ≥ 2, N (z)dz m N (z) a.s.

Table des matières

 des

	1 Introduction

  . Suivant les travaux de Mestre, la première partie dans le Chapitre 4 établit un CLT pour son estimateur dans le cas gaussien. La méthode consiste à établir le CLT pour Tr(Y * n

  Section 4.1.5). 3. L'estimation par la méthode des moments pose en général un problème de simulation.Le rapport est organisé de la façon suivante.-le Chapitre 2 établit un CLT de statistiques linéaires pour les fonctionnelles analytiques par la méthode classique sur les martingales. Nous exprimerons la covariance sous une seule formule dans le cas réel ou complexe. Cela fait l'objet du papier J. Yao et J. Najim. "Fluctuations for linear spectral statistics of large information-plusnoise type random matrices", en préparation.-Le Chapitre 3 étudit le CLT pour les fonctions non-analytiques en utilisant la transformée de Fourier. Nous préparons le papier J. Yao et J. Najim. "Gaussian fluctuations for linear spectral statistics of large random covariance matrices", en préparation.

Trouver un bon algorithme pour assurer la bonne performance de simulation reste un problème ouvert. (cf. Section 4.2.4) 1.4 Organisation du rapport -Dans le Chapitre 4, nous allons étudier les fluctuations de l'estimateur de Mestre dans le cas gaussien. Une application sur radio cognitive sera aussi proposée. La seconde partie du Chapitre 4 construit un estimateur consistant sans " condition de séparabilité". Les fluctuations seront également étudiées. La preuve mélange la méthode par l'intégrale de contours et la méthode des moments. C'est l'objet des articles J. Yao, R. Couillet, J. Najim, and M. Debbah. "Fluctuations of an improved population Chapitre 2

  .1.1) and by m YnY * n (z) and m Y * n Yn (z) their normalized trace, which are respectively the Stieltjes transforms of the empirical distribution of Y n Y * n 's eigenvalues and of Y *

n Y n 's eigenvalues :

  the largest eigenvalue of Y n Y * n will eventually lie below ℓ + and representation (2.1.5) will eventually hold true for a contour Γ containing [0, ℓ + ]. Assumption 2.2.3 may be slightly strengthened, but in this case estimates over the extreme eigenvalues of Y n Y *

n (see for instance Lemma 2.3.4 below) would be more challenging to obtain.

  2.1 while the second one yields a bias and is handled in Theorem 2.2.2. Let Assumptions 2.2.1 and 2.2.2 hold true ; let k be a given integer and f = (f i ; 1 ≤ i ≤ k) where the f i 's are functions satisfying Assumption 2.2.3.

	Theorem 2.2.1 Consider the random
	vector
	L 1 n (f )

  Theorem 2.2.2 Let Assumptions 2.2.1 and 2.2.2 hold true ; assume that f is a function satis-

	fying Assumption 2.2.3. Then
	L 2 n

  Proposition 2.3.1 (Truncation of the entries) Let Assumptions 2.2.1 and 2.2.2 hold true and let f satisfy Assumption 2.2.3, then there exists a sequence η n ↓ 0 with η n n 1/5 → ∞ such that if Xn is the N × n matrix with centered and normalized entries

  TrM| p ≤ K p ((E|x 1 | 4 TrMM * ) p/2 + E|x 1 | 2p Tr(MM * ) p/2 ).

	Another consequence of Lemma 2.3.1 is the following lemma.
	Lemma 2.3.3 Under the same setting as Lemma 2.3.2, we have
		E y * n M n y n -	TrM n n	-a * n M n a n	4	≤	Kη 4 n n	,
	where η n is defined in Proposition 2.3.1.
	Proofs of Lemma 2.3.2 and 2.3.3 are postponed to Appendices 2.6.3 and 2.6.4.
													.3.15)
	Lemma 2.3.1 [9, Lemma 2.7] Let x = (x 1 , • • • , x n ) be an n×1 vector where x i are centered i.i.d. complex random variables with unit variance and bounded fourth moment. Let M be a Hermitian
	complex matrix. Then for any p ≥ 2, there exists a constant K p depending only on p for which
	E|x A direct consequence of this lemma is the following proposition.
	Lemma 2.3.2 [11, Eq. (3.2)] Let x n be a sequence of random vectors as in the statement of
	the precedent lemma and M n be a sequence of n × n matrices independent of x n with uniformly bounded spectral norm. Denote by y n = 1 √ n x n + a n where a n is a deterministic vector with
	uniformly bounded Euclidian norm. Then
	max E	x * n M n x n n	-	TrM n n	2	, E y * n M n y n -	TrM n n	-a * n M n a n	2	≤	K n	.

* Mx -

  1} n,N →∞ forms a tight sequence over C N , hence relatively compact in distribution. As the distribution L(N n ) of the Gaussian process N n is the limit in M of the distribution of M n,M , belongs to the closure of {M n,M : M ≥ 1} n,N →∞ , which is relatively compact. Finally, L(N n ) is included in a compact set, hence is relatively compact. In particular, the family of gaussian processes (N n ) is tight. This proves that the variance Θ n (z) is uniformly bounded.

				which satisfies Assumption 2.2.2}.
	Since A M n	= A n for all M ≥ 1, we have
		sup M ≥1,N,n→∞	A M n	= sup N,n→∞	A n < ∞
	by Assumption 2.2.2. The arguments with Billingsley's condition (cf. (2.3.28)) show that the
	family {M n,M : M ≥

  We first prove that|A n (z 1 , z 2 )| and |L n (z 1 , z 2 )| are bounded. Recall v i = |Imz i | for i = 1, 2.

	2 <
	∞ which is done in (2.3.27). This proves (2.4.21). Now we will prove (2.4.22) based on the following lemma.
	Lemma 2.4.1 [11, Lemma 2.3] Let (f n ) n≥1 be analytic in D, a connected open set of E, sa-
	tisfying |f n (z)| ≤ M for every n and z in D, and f n (z) converges, as n → ∞ for each z in a subset of D having a limit point in D. Then there exists a function f , analytic in D for which
	f n (z) → f (z) and f ′ n (z) → f ′ (z) for all z ∈ D. Moreover, on any set bounded by a contour interior to D the convergence is uniform and f n (z) is uniformly bounded by 2M/ε, where ε is
	the distance between the contour and the boundary of D.
	Using (2.3.24) and Lemma 2.3.2, we have

  .4.24) Remark 2.4.2 z 1 , z 2 ∈ C n are fixed in this section. Then |z i | and |Imz i | are bounded for i = 1, 2. We shall not mention the dependence on |z i | or |Imz i | in the sequel. From (2.4.17) and (2.4.18) in Theorem 2.4.1, E| bjtjj | 2 ≤ 2E| bjqjj | 2 +2E|q jj -tjj | 2 ≤ K/n. Therefore, by Cauchy-Schwarz and Lemma 2.3.3,

  the expression of ζ M kj (cf. (2.4.49)) with ζ M kj (cf. (2.4.31)), we get

  ). By Proposition 2.4.2, and with the same argument as Lemma 2.4.4, the terms with E[|x 1 | 2 x 1 ] and E[|x 1 | 2 x1 ] will not contribute in the sum. Then we have

  1/2 ] kj and [Y T U (t)Y ] kk .

	Lemma 3.2.2 We have :

  1/2 ] jj . Remark 3.2.4 As U(t, s) = e itYsY * s , when we derive U(t, s) according to real Gaussian variable, both Y s and Y * s depend on Xr,jk . However, Y s does not depend on Xc,jk .

  and B 1,n , B 2,n being defined in (3.1.10) and (3.1.11) respectively.

	Then we have
	lim sup n

Here F M denotes the empirical distribution of matrix M's eigenvalues.

For two complex Hermitian matrices A and B, we denote A ≤ B if B -A is a Hermitian nonnegative matrix.

By cluster, we mean a connected component of the support of the limiting probability distribution of the spectrum.

Remerciements

Consider the random vector

where N 1 n (f ) is defined in (3.1.4). Define the Gaussian random vector

with zero mean and the covariance

where Θ n = 2Θ 0,n + κ 4 Θ 1,n and Θ 0,n , Θ 1,n are defined in (3.3.1) and (3.3.2) respectively. The contours here are assumed to be non-overlapping and closed, taken in the positive direction in the complex plane and enclosing the interval (3.3.3).

Then, the sequence of R k -valued random variable Z 1 n (f ) is tight and the random vector N 1 n (f ) satisfies :

Outline of the proof : By Fourier transform, as f is integrable in R, for x ∈ R,

and by the inverse formula, we have

Then

Ee iλ k t dt.

It is natural to study the fluctuation of the process ů(t) = N k=1 e iλ k t -N k=1 Ee iλ k t . We will verify all conditions in Lemma 3.1.2. In particular, the condition

2 is ensure by Theorem 3.1.3. Then the expression of of the variance will be investigated.

Convergence of the process N 1 n (f )

We start the proof with the convergence of u n (x). In particular, we will show that all conditions in Lemma 3.1.2 are fulfilled.

λ j -z where λ i are eigenvalues of YY T and M 1 n (z) =

where v = t(z) t(z).

For each matrix M, we define ImM = 1 2i (M -M * ). Noticing that T * (z) = T(z) and    T T = (-z(I + t(z) R)) -1 T * = (-z(I + t(z)R)) -1 we have :

Taking trace and divided by n, we have

where ṽ = 1 n Tr[RT(z)T * (z)R]. Thus we get :

where D(z) is defined in (3.4.20). This is equivalent to On the other hand, Im( t(z)) can be written as

We recall that in [START_REF] Najim | Gaussian fluctuations for linear spectral statistics of large random covariance matrices[END_REF], it is shown that the sequence Fn is tight. This implies that it exists a constant C > 0 for which

(3.5.9) Plugging (3.5.8) and (3.5.9) into (3.5.7), we get

We now establish (3.4.25). Recall the definition of S and τ in (3.4.5) and (3.4.6) respectively. We express Im(E mc (z)) and 1 n Im(zETrRQ) as

,

Then by calculating the imaginary parts of τ (z) and z n TrRS, we have

where v ′ and ṽ′ are defined in (3.4.23). Then

where D ′ is defined in (3.4.21). We get the system .5.11) This equation is similar to (3.5.6) with two extra error terms 1 n Im[ε(z)] and 1 n Im(z ε(z)). However,

Therefore, ξ n converges weakly to Fn and there exists an integer N 1 and C>0 such that for n ≥ N 1 ,

With the same argument as (3.5.9), we obtain

(3.5.12)

Chapitre 4

Estimation for large dimensional random matrices

Fluctuations of an improved estimator in sample covariance matrix models

The content below is evolved from the papers [START_REF] Yao | Clt for eigen-inference methods in cognitive radios[END_REF][START_REF] Yao | Fluctuations of an improved population eigenvalue estimator in sample covariance matrix models[END_REF]. Problems of statistical inference based on M independent observations of an N -variate random variable y, with E[y] = 0 and E[yy H ] = R N have drawn the attention of researchers from many fields for years : Portfolio optimization in finance [START_REF] Plerous | Random matrix approach to cross correlations in financial data[END_REF], gene coexistence in biostatistics [START_REF] Luo | Constructing gene co-expression networks and predicting functions of unknown genes by random matrix theory[END_REF], channel capacity in wireless communications [START_REF] Kammoun | BER and Outage Probability Approximations for LMMSE Detectors on Correlated MIMO channels[END_REF], power estimation in sensor networks [START_REF] Couillet | Eigen-inference for energy estimation of multiple sources[END_REF], distance of targets in array processing [START_REF] Mestre | Modified Subspace Algorithms for DoA Estimation With Large Arrays[END_REF], etc.

In particular, retrieving spectral properties of the population covariance matrix R N , based on the observation of M independent and identically distributed (i.i.d.) samples y (1) , . . . , y (M ) , is paramount to many questions of general science. If M is large compared to N , then it is known that almost surely RN -R N → 0, as M → ∞, for any standard matrix norm, where RN is the sample covariance matrix RN 1 M M m=1 y (m) y (m)H . However, one cannot always afford a large number of samples. In order to cope with this issue, random matrix theory [START_REF] Bai | Spectral Analysis of Large Dimensional Random Matrices[END_REF][START_REF] Couillet | Random matrix methods for wireless communications[END_REF] has proposed new estimators, mainly spurred by the G-estimators of Girko [START_REF] Girko | Ten years of general statistical analysis[END_REF]. Other works include convex optimization methods [START_REF] Silverstein | Signal detection via spectral theory of large dimensional random matrices[END_REF][START_REF] Karoui | Spectrum estimation for large dimensional covariance matrices using random matrix theory[END_REF], free probability tools [START_REF] Ryan | Free deconvolution for signal processing applications[END_REF][START_REF] Couillet | Free deconvolution for OFDM multicell SNR detection[END_REF], and regularized estimation (banding, tapering, thresholding, etc.) [START_REF] Bickel | Regularized estimation of large covariance matrices[END_REF][START_REF] Bickel | Regularized estimation of large covariance matrices[END_REF][START_REF] Cai | Optimal rates of convergence for covariance matrix estimation[END_REF], when the structure of R N is known. Many of those estimators are consistent in the sense that they are asymptotically unbiased as M, N grow large at the same rate. Nonetheless, only recently have techniques been unveiled which allow to estimate individual eigenvalues and functionals of eigenvectors of R N . The first contributor is Mestre [START_REF] Mestre | On the asymptotic behavior of the sample estimates of eigenvalues and eigenvectors of covariance matrices[END_REF]- [START_REF] Mestre | Improved estimation of eigenvalues of covariance matrices and their associated subspaces using their sample estimates[END_REF] who studied the case where RN = R

N with R N having eigenvalues with large multiplicities and unknown eigenvectors, and U N with i.i.d. entries. For this model, he provides an estimator for every eigenvalue of R N with large multiplicity under some separability condition, see also Vallet et al. [92], Couillet et al. [START_REF] Couillet | Eigen-inference for energy estimation of multiple sources[END_REF] for more elaborate models.

These estimators, although proven asymptotically unbiased, have nonetheless not been fully characterized in terms of their asymptotic performances. It is in particular fundamental to eva-where

if R k and R ′ k are two contours which only contain the k-th cluster. This fact will be of importance later.

The main idea of the proof of the theorem lies in three steps :

(i) To prove the convergence in distribution of the process (X N , X ′ N , u N , u ′ N ) to a Gaussian process.

(ii) To transfer this convergence to the quantity

with the help of the continuous mapping theorem [START_REF] Kallenberg | Foundations of mordern Probability[END_REF].

Remark 4.1.6 Note that the convergence in step (i) is a distribution convergence at a process level, hence one has to first establish the finite dimensional convergence of the process and then to prove that the process is tight over C k (see for instance [START_REF] Billingsley | Probability and Measure[END_REF]Theorem 13.1]). Tightness turns out to be difficult to establish due to the lack of control over the eigenvalues of RN whenever the contour crosses the real line. In order to circumvent this issue, we shall introduce, following Bai and Silverstein [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF], a process that approximates X N and X ′ N .

Let us now start the proof of Theorem 4.1.3. We begin by simple considerations on complex Gaussian random vectors. Consider a C 2valued, centered, random vector (U, V ). If (U, V ) is, as an R 4 -valued vector, Gaussian, then its distribution is fully characterized with the quantities :

is continuous and admits partial derivatives up to order 2 over (C \ S) 2 .

In a decentralized context, where each user selects its own power from this finite set according to a defined control energy strategy, the base station which stands for the receiver can have to estimate the number of users in each class as well as their corresponding powers.

Obviously, this problem amounts to estimating the eigenvalues of the theoretical covariance matrix as well as their corresponding multiplicities. Similar scenarios are studied in [START_REF] Yao | Clt for eigen-inference methods in cognitive radios[END_REF][START_REF] Yao | Fluctuations of an improved population eigenvalue estimator in sample covariance matrix models[END_REF]. Among the proposed parametric techniques, we cite the one developed by Mestre [START_REF] Mestre | Improved estimation of eigenvalues of covariance matrices and their associated subspaces using their sample estimates[END_REF] and taken up by Vallet et al [START_REF] Vallet | Improved subspace estimation for multivariate observations of high dimension : The deterministic signals case[END_REF] and Couillet et al [START_REF] Couillet | Eigen-inference for energy estimation of multiple sources[END_REF] for more elaborated models. Although being computationally efficient, this technique requires a separability condition, namely the assumption that the number of samples is large compared to the dimension of each sample (small limiting ratio c = lim N M > 0). In such a case, the limiting spectrum of the empirical covariance matrix possesses as many clusters 3 as there are eigenvalues to be estimated, and each eigenvalue can be estimated by a contour integral surrounding the related cluster. Mestre's technique cannot be applied any more in the case where c is larger (which reflects a higher dimension of the observations relatively to the sample dimension). In fact, the dimension of the clusters may grow and neighbouring clusters may merge, violating the one-to-one correspondence between clusters and eigenvalues to be estimated (see for instance Fig. 4.1 and 4.2).

A way to circumvent the separability condition has recently been proposed by Bai, Chen and Yao [START_REF] Bai | On estimation of the population spectral distribution from a high-dimensional sample covariance matrix[END_REF], based on the use of the empirical asymptotic moments :

which can be shown to be a sufficient statistic to estimate

Although being robust to separability condition, this technique suffers from numerical difficulties, since the proposed estimator has no closed-form expression and thus should be determined numerically. An interesting contribution, although not directly focused on estimating the covariance of the observations is the work by Rubio and Mestre [START_REF] Rubio | Generalized Consistent Estimation on Low-rank Krylov Subspaces of Arbitrarily High Dimension[END_REF], where an alternative way to estimate the moments

for all k ∈ N is proposed, yielding an explicit (yet lengthy) formula.

In this chapter, we improve existing work in several directions : With respect to Mestre's seminal papers [START_REF] Mestre | On the asymptotic behavior of the sample estimates of eigenvalues and eigenvectors of covariance matrices[END_REF][START_REF] Mestre | Improved estimation of eigenvalues of covariance matrices and their associated subspaces using their sample estimates[END_REF], we propose a joint estimation of the eigenvalues and their multiplicities, and drop the separability assumption. The proposed estimator is close in spirit to the one developed by Bai et al. in [START_REF] Bai | On estimation of the population spectral distribution from a high-dimensional sample covariance matrix[END_REF], although we carefully establish the existence and uniqueness of the estimator, which is not explicit in [START_REF] Bai | On estimation of the population spectral distribution from a high-dimensional sample covariance matrix[END_REF]. Comparisons on the relative numerical efficiency of both procedures is provided in the simulations section. Finally, we study the fluctuations of the estimator and establish a Central Limit Theorem (CLT).

The remainder of this chapter is organized as follows. In Section 4.2.1, the main assumptions are provided and Mestre's estimator [START_REF] Mestre | Improved estimation of eigenvalues of covariance matrices and their associated subspaces using their sample estimates[END_REF] is briefly reviewed. In Section 4.2.2, the proposed estimator is described. Its fluctuations are studied in Section 4.2.3, where a CLT is stated. Si-

Estimation of the eigenvalues ρ i

In this section, we provide a method to estimate consistently the eigenvalues of the population covariance matrix and their multiplicities without the need of the separability condition (cf. Fig. 4.2). Our method is based on the asymptotic evaluation of the moments of the eigenvalues of

If ( m i ) 1≤i≤2L-1 are the empirical moments of the sample eigenvalues, then it is well-known that except for i = 1, γ i cannot be approximated by m i . Consistent estimators for γ i are provided in [START_REF] Rubio | Generalized Consistent Estimation on Low-rank Krylov Subspaces of Arbitrarily High Dimension[END_REF], where it has been proved that :

where

µ S (l, i) being some given coefficients that depend on the system dimensions and on the empirical moments m i [START_REF] Rubio | Generalized Consistent Estimation on Low-rank Krylov Subspaces of Arbitrarily High Dimension[END_REF]

where C is a counterclockwise oriented contour which encloses the support S of the limiting distribution of the eigenvalues of RN . Let γ i be the theoretical moments as given in (4.2.2). Then, for

The proof of this lemma is postponed to Appendix 4.3.4. While the estimates proposed by [START_REF] Rubio | Generalized Consistent Estimation on Low-rank Krylov Subspaces of Arbitrarily High Dimension[END_REF] are better in practice, estimates (γ i ) will be of interest in order to establish the Central Limit Theorem, and to obtain a closed-form expression of the asymptotic variance. An interesting remark is that the map that links the eigenvalues and their multiplicities to their first 2L -1 moments is invertible. Retrieving the eigenvalues from the estimates of the 2L -1 moments is thus possible. This is the basic idea on which our method is founded.

The main result is stated as below :

On the other hand, we have Γ = ADA T , where

Then,

Therefore, the vector s is then uniquely determined by Γ and b and is given by :

Hence the uniqueness. Proof of Theorem 4.2.2 is completed.

Summary of the main steps of the estimation procedure

The proof of the uniqueness shows that the solutions of the system of equations (4.2.4) can be directly obtained from the estimates of the first 2L -1 moments. More precisely, the estimation of the eigenvalues and their corresponding multiplicities can be performed through the following steps :

N, M

Proposed method Bai, Chen and Yao's method N = 300, M = 800 0.5s 10.68s N = 360, M = 960 0.55s 25.57s N = 420, M = 1120 0.67s 42.62s

Table 4.1 -Execution time to obtain an estimator for one realization (in seconds)

Accuracy of the Gaussian approximation

Finally, we verify by simulations the accuracy of the Gaussian approximation. We consider the case where there are two different eigenvalues ρ 1 = 1 and ρ 2 = 3 that are uniformly distributed. Unlike the first experiment, we assume that the multiplicities are unknown. We represent in Fig 4 .6 the histogram for ρ1 and ρ2 when N = 60 and M = 120. We also represent in red line, the corresponding Gaussian distribution. We note that as it was predicted by our derived results, the histogram is similar to that of a Gaussian random variable. Let us first begin by considerations related to the supports of the probability distributions associated to m(z) and m N (z). Denote by S and S N these supports and recall that S is the union of L disjoint clusters : For

Appendices

by replacing N M by c and F R N by F R . As N M → c and K i M → c i , the roots of Eq. (4.3.3) converge to those of the limiting equation (see [START_REF] Cucker | An Alternate Proof of the Continuity of the Roots of a Polynomial[END_REF] for instance). Hence the conclusion.

We are now in position to establish the proof of Proposition 4.1.1.

Denote by S(ε) the ε-blow-up of S, i.e. S(ε) = {x ∈ R, d(x, S) < ε}. Let ε > 0 be small enough and consider a smooth function φ equal to zero on S(ε/3), equal to 1 if x / ∈ S(ε), equal to zero again if |x| ≥ τ (as we shall see, τ will be chosen to be large), and smooth in-between with 0 ≤ φ ≤ 1 :

Notice that if N, M → ∞ and N is large enough, then by Lemma 4.3.1, φ(x) = 0 for all x ∈ S N . Now if Z is a M ×M hermitian matrix with spectral decomposition

where U is unitary and

We have : We shall therefore establish estimates over E[Tr φ( RN )] p . Take p = 2 k ; we prove the following statement by induction : For k ≥ 1 and for every integer β < 2 k and for every smooth function f with compact support whose value on S(ε/3) is zero , E Trf ( RN )

First note that, due to Lemma 4. Let k > 0 be fixed and assume that the result holds true for β < 2 k . We want to show that

). At step k + 1, the expectation writes :

This partition of Rk is identical to that used in [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF]Section 1]. With probability one (see [START_REF] Bai | No Eigenvalues Outside the Support of the Limiting Spectral Distribution of Large Dimensional Sample Covariance Matrices[END_REF] and [START_REF] Bai | Exact Separation of Eigenvalues of Large Dimensional Sample Covariance Matrices[END_REF]), for all ǫ > 0, lim

with d(x, S) the Euclidean distance of x to the set S. Notice that :

Furthermore, with probability one, for all N large,

and

Thus, with probability one,

where K 1 is a constant which does not depend on N, M . A similar result can be achieved for the derivative functions X ′ N (z) and X′ N (z). One can get :

With probability one, for all N, M large,

for some constants K 1 and K 2 . Both terms converge to zero as N, M → ∞. Then, by Slutsky's lemma [START_REF] Vaart | Asymptotic statistics[END_REF], it suffices to establish the arguments for XN (z) and X′ N (z). As mentioned in Section 4.1.4, there are two conditions to prove (see for instance Billingsley [START_REF] Billingsley | Probability and Measure[END_REF]Theorem 13.1]) to establish the convergence in distribution of the process ( XN , X′ N ) to the process (X, Y ) over the compact K :

-Finite-dimensional convergence of the process ( XN , X′ N ) over the compact K. -Tightness on the compact K.

Finite-dimensional convergence

In [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF], Bai and Silverstein establish a central limit theorem for F RN with the complex Gaussian entries X ij . We recall below their main result. Proposition 4.3.1 (cf. [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF]) With the notations introduced in Section 4.1.1, for f 1 , . . . , f p , analytic on an open region containing R,

where V = (V ij ) and

where the integration is over positively oriented contours C 1 and C 2 which are supposed to be non-overlapping and both circle around the support S. Now we apply Proposition 4.3.1 to establish the finite-dimensional convergence. For all z i ∈ K\R, note that

with the contour C which contains the support S. As X N (z) = M (m RN (z)-m N (z)), Proposition 4.3.1 directly implies that for every finite p ∈ N, the random vector

converges to a centered Gaussian vector by considering the functions :

where for all

Hence the finite dimensional convergence.

The proof of the tightness is based on Poincaré-Nash inequality (see for instance [START_REF] Vallet | Improved subspace estimation for multivariate observations of high dimension : The deterministic signals case[END_REF] and [START_REF] Hachem | A new approach for capacity analysis of large dimensional multi-antenna channels[END_REF]). In Appendix 4.3.1, it is proved that for all ǫ > 0 and all ℓ ∈ N, P sup

Following the same idea as Bai and Silverstein [START_REF] Bai | CLT of linear spectral statistics of large dimensional sample covariance matrices[END_REF]Section 3 and 4], it is indeed a tight sequence. The details of the proof are in Appendix 4.3.2. Thus Lemma 4.1.1 is proved.

Tightness

We will show the tightness of the sequence M (m RN m N ) and M (m

) by using Poincaré-Nash's inequality [START_REF] Hachem | A new approach for capacity analysis of large dimensional multi-antenna channels[END_REF] on the compact K. As the compact K is the union of 2L contours R k and R ′ ℓ , it is sufficient to prove the tightness on every contour R k (or equivalently Rk ). First, denote by

). We now prove tightness based on [START_REF] Billingsley | Probability and Measure[END_REF]Theorem 13.1], i.e.

1. Tightness at any point of the contour (here Rk ).

Satisfaction of the condition

Condition 1) is achieved by an immediate application of Proposition 4.3.1. We now verify the second condition.

We evaluate

The next step is to prove the tightness of M (m ′ RN (z)m ′ N (z)). We have

Following the same method as derived before, one obtains

Then Poincaré-Nash inequality yields that N (z) is achieved as before. The proof of the tightness is completed if one verifies that X2 ′ N (z) for z ∈ Rk is bounded and forms an equicontinuous family, and converges to 0. We will use the same method for the process X2 N (z) (see [START_REF] Bai | Spectral Analysis of Large Dimensional Random Matrices[END_REF]Section 9.11]).

By Formula (9.11.1) in [START_REF] Bai | Spectral Analysis of Large Dimensional Random Matrices[END_REF]Section 9.11], it is proved that :

where

If one differentiates (4.3.9) with respect to z, the equation becomes

In the work of [5, Section 9.11], it is proved that when N, M tend to infinity,

With these results, it suffices to show that T ′ N → 0, and X2 ′ N is equicontinuous. In [5, Section 9.9], they show that for m, p ∈ N and a non-random N × N matrix A k , k = 1, .., m and B ℓ , ℓ = 1, .., q, we have

We have also that for any positive p,

where K p is a constant which depends only on p. With all these preliminaries, as T N → 0, by the dominated convergence theorem of derivation, it suffices to show that T ′ N is bounded over Rk . In [START_REF] Bai | Spectral Analysis of Large Dimensional Random Matrices[END_REF]Section 9.11], it is sufficient to show that (f ′ M (z)) is bounded where

With the help of (

with certain k 1 , k 2 , k 3 , k 4 , k 5 ∈ N. Thanks to (4.3.11) and (4.3.12), these terms are all bounded. Therefore, RHS is indeed bounded. This ends the proof of the tightness.

Study of the set R z 2

For z 2 fixed, denote R z 2 = {z 1 ∈ C : z 1 = z 2 , m RN (z 1 ) = m RN (z 2 )}. We will show that this set is empty a.s. for all N, M large. Suppose that z 1 ∈ R z 2 . We first use [START_REF] Bai | Spectral Analysis of Large Dimensional Random Matrices[END_REF]Formula (9.11.4)] that sup By Formula (4.3.1), we get

As z 1 = z 2 , m N (z 1 ) = m N (z 2 ). --→ 0 that for all large N, M , R z 2 = ∅ almost surely.

Proof of lemma 4.2.1

By Cauchy's formula, write :

where Γ is a counterclockwise oriented contour that surrounds all the eigenvalues {ρ 1 , • • • , ρ L }.

Performing the changing variable ω = -1 m N (z) in the same manner as in [START_REF] Mestre | Improved estimation of eigenvalues of covariance matrices and their associated subspaces using their sample estimates[END_REF], we get :