domain: info math spi

text:
FRANC ¸OIS CHAPON Romain Couillet

WALID Xavier Mestre

THE OUTLIERS AMONG THE SINGULAR VALUES OF LARGE RECTANGULAR RANDOM MATRICES WITH ADDITIVE FIXED RANK DEFORMATION

Keywords: 2000 Mathematics Subject Classification. Primary 15A52, Secondary 15A18, 60F15 Random Matrix Theory, Stieltjes Transform, Fixed rank deformation, Extreme eigenvalues, Gaussian fluctuations

Consider the matrix Σn = n -1/2 XnD 1/2 n + Pn where the matrix Xn ∈ C N×n has Gaussian standard independent elements, Dn is a deterministic diagonal nonnegative matrix, and Pn is a deterministic matrix with fixed rank. Under some known conditions, the spectral measures of ΣnΣ * n and n -1 XnDnX * n both converge towards a compactly supported probability measure µ as N, n → ∞ with N/n → c > 0. In this paper, it is proved that finitely many eigenvalues of ΣnΣ * n may stay away from the support of µ in the large dimensional regime. The existence and locations of these outliers in any connected component of Rsupp(µ) are studied. The fluctuations of the largest outliers of ΣnΣ * n are also analyzed. The results find applications in the fields of signal processing and radio communications.

where X n is a N × n random matrix whose coefficients X ij are independent and identically distributed (iid) complex Gaussian random variables such that ℜ(X 11 ) and ℑ(X 11 ) are independent, each with mean zero and variance 1/2, and where D n is a deterministic nonnegative diagonal n × n matrix. Writing D n = diag(d n j ) j=1,...,n and denoting by δ the Dirac measure, it is assumed that the spectral measure ν n = n -1 n j=1 δ d n j of D n converges weakly to a compactly supported probability measure ν when n → ∞. It is also assumed that the maximum of the distances from the diagonal elements of D n to the support supp(ν) of ν goes to zero as n → ∞. Assume that N/n → c when n → ∞, where c is a positive constant. Then it is known that with probability one, the spectral measure of the Gram matrix n -1 Y n Y * n converges weakly to a compactly supported probability measure µ (see [START_REF] Marčenko | Distribution of eigenvalues in certain sets of random matrices[END_REF], [START_REF] Girko | of Mathematics and its Applications[END_REF], [START_REF] Silverstein | Strong convergence of the empirical distribution of eigenvalues of largedimensional random matrices[END_REF], [START_REF] Silverstein | On the empirical distribution of eigenvalues of a class of large-dimensional random matrices[END_REF]) and, with probability one, n -1 Y n Y * n has no eigenvalues in any compact interval outside supp(µ) for large n [START_REF] Bai | No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices[END_REF]. Let r be a given positive integer and consider a sequence of deterministic N × n matrices P n , n = 1, 2, . . ., such that rank(P n ) = r and sup n P n < ∞ where • is the spectral norm. Consider the matrix Σ n = n -1/2 Y n + P n . Since the additive deformation P n has a fixed rank, the spectral measure of Σ n Σ * n still converges to µ (see, e.g., [START_REF] Bai | Methodologies in spectral analysis of large-dimensional random matrices, a review[END_REF]Lemma 2.2]). However, a finite number of eigenvalues of Σ n Σ * n (often called "outliers" in similar contexts) may stay away of the support of µ. In this paper, minimal conditions ensuring the existence and the convergence of these outliers towards constant values outside supp(µ) are provided, and these limit values are characterized. The fluctuations of the outliers lying at the right of supp(µ) are also studied.

The behavior of the outliers in the spectrum of large random matrices has aroused an important research effort. In the statistics literature, one of the first contributions to deal with this subject was [START_REF] Johnstone | On the distribution of the largest eigenvalue in principal components analysis[END_REF]. It raised the question of the behavior of the extreme eigenvalues of a sample covariance matrix when the population covariance matrix has all but finitely many of its eigenvalues equal to one (leading to a mutliplicative fixed rank deformation). This problem has been studied thoroughly in [START_REF] Baik | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices[END_REF][START_REF] Baik | Eigenvalues of large sample covariance matrices of spiked population models[END_REF][START_REF] Paul | Asymptotics of sample eigenstructure for a large dimensional spiked covariance model[END_REF]. Other contributions (see [START_REF] Capitaine | The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations[END_REF]) study the outliers of a Wigner matrix subject to an additive fixed rank deformation. The asymptotic fluctuations of the outliers have been addressed in [START_REF] Baik | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices[END_REF][START_REF] Péché | The largest eigenvalue of small rank perturbations of Hermitian random matrices[END_REF][START_REF] Paul | Asymptotics of sample eigenstructure for a large dimensional spiked covariance model[END_REF][START_REF] Bai | Central limit theorems for eigenvalues in a spiked population model[END_REF][START_REF] Capitaine | The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations[END_REF][START_REF] Capitaine | Central limit theorems for eigenvalues of deformations of Wigner matrices[END_REF][START_REF] Benaych-Georges | Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices[END_REF].

Recently, Benaych-Georges and Nadakuditi proposed in [START_REF] Benaych-Georges | The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices[END_REF][START_REF] Benaych-Georges | The singular values and vectors of low rank perturbations of large rectangular random matrices[END_REF] a generic method for characterizing the behavior of the outliers for a large palette of random matrix models. For our model, this method shows that the limiting locations as well as the fluctuations of the outliers are intimately related to the asymptotic behavior of certain bilinear forms involving the resolvents (n

-1 Y n Y * n -xI N ) -1 and (n -1 Y * n Y n -xI n ) -1
of the undeformed matrix for real values of x. When D n = I n , the asymptotic behavior of these bilinear forms can be simply identified (see [START_REF] Benaych-Georges | The singular values and vectors of low rank perturbations of large rectangular random matrices[END_REF]) thanks to the fact that the probability law of Y n is invariant by left or right multiplication by deterministic unitary matrices. For general D n , other tools need to be used. In this paper, these bilinear forms are studied with the help of an integration by parts formula for functionals of Gaussian vectors and the Poincaré-Nash inequality. These tools belong to the arsenal of random matrix theory, as shown in the recent monograph [START_REF] Pastur | Eigenvalue distribution of large random matrices[END_REF] and in the references therein. In order to be able to use them in our context, we make use of a regularizing function ensuring that the moments of the bilinear forms exist for certain x ∈ R + = [0, ∞).

The study of the spectrum outliers of large random matrices has a wide range of applications. These include communication theory [START_REF] Hachem | A subspace estimator for fixed rank perturbations of large random matrices[END_REF], fault diagnosis in complex systems [START_REF] Couillet | Fluctuations of spiked random matrix models and failure diagnosis in sensor networks[END_REF], financial portfolio management [START_REF] Potters | Financial applications of random matrix theory: Old laces and new pieces[END_REF], or chemometrics [START_REF] Naes | A User-Friendly Guide to Multivariate Calibration and Classification[END_REF]. The matrix model considered in this paper is widely used in the fields of multidimensional signal processing and radio communications. Using the invariance of the probability law of X n by multiplication by a constant unitary matrix, D n can be straightforwardly replaced with a nonnegative Hermitian matrix R n . In the model

X n R 1/2 n + P n where R 1/2 n
is any square root of R n , matrix P n often represents n snapshots of a discrete time radio signal sent by r sources and received by an array of N antennas, while X n R 1/2 n is a temporally correlated and spatially independent "noise" (spatially correlated and temporally independent noises can be considered as well). In this framework, the results of this paper can be used for detecting the signal sources, estimating their powers, or determining their directions. These subjects are explored in the applicative paper [START_REF] Vinogradova | Statistical inference in large antenna arrays under unknown noise pattern[END_REF].

The remainder of the article is organized as follows. The assumptions and the main results are provided in Section 2. The general approach as well as the basic mathematical tools needed for the proofs are provided in Secion 3. These proofs are given in Sections 4 and 5, which concern respectively the first order (convergence) and the second order (fluctuations) behavior of the outliers.

Problem description and main results

Given a sequence of integers N = N (n), n = 1, 2, . . ., we consider the sequence of N × n matrices Σ n = n -1/2 Y n + P n = n -1/2 X n D 1/2 n + P n with the following assumptions:

Assumption 1. The ratio c n = N (n)/n converges to a positive constant c as n → ∞.

Assumption 2. The matrix X n = [X ij ] N,n i,j=1 is a N × n random matrix whose coefficients X ij are iid complex random variables such that ℜ(X 11 ) and ℑ(X 11 ) are independent, each with probability distribution N (0, 1/2). Assumption 3. The sequence of n×n deterministic diagonal nonnegative matrices D n = diag(d n j ) n j=1 satisfies the following: (1) The probability measure ν n = n -1 n j=1 δ d n j converges weakly to a probability measure ν with compact support. The asymptotic behavior of the spectral measure of n -1 Y n Y * n under these assumptions has been thoroughly studied in the literature. Before pursuing, we recall the main results which describe this behavior. These results are built around the Stieltjes Transform, defined, for a positive finite measure µ over the Borel sets of R, as

m(z) = R 1 t -z µ(dt) (1)
analytic on Csupp(µ). It is straightforward to check that ℑm(z) ≥ 0 when z ∈ C + = {z : ℑ(z) > 0}, and sup y>0 |ym(ıy)| < ∞. Conversely, any analytic function m(z) on C + that has these two properties admits the integral representation [START_REF] Bai | Central limit theorems for eigenvalues in a spiked population model[END_REF] where µ is a positive finite measure. Furthermore, for any continuous real function ϕ with compact support in R,

ϕ(t) µ(dt) = 1 π lim y↓0 ϕ(x)ℑm(x + ıy) dx (2)
which implies that the measure µ is uniquely defined by its Stieltjes Transform. Finally, if ℑ(zm(z)) ≥ 0 when z ∈ C + , then µ((-∞, 0)) = 0 [START_REF] Kreȋn | The Markov moment problem and extremal problems[END_REF]. These facts can be generalized to Hermitian matrix-valued nonnegative finite measures [START_REF] Bolotnikov | On a general moment problem on the half axis[END_REF][START_REF] Gesztesy | On matrix-valued Herglotz functions[END_REF]. Let m(z) be a C r×r -valued analytic function on z ∈ C + . Letting ℑX = (X -X * )/(2ı), assume that ℑm(z) ≥ 0 and ℑ(zm(z)) ≥ 0 in the order of the Hermitian matrices for any z ∈ C + , and that sup y>0 ym(ıy) < ∞.

Then m(z) admits the representation (1) where µ is now a r × r matrix-valued nonnegative finite measure such that µ((-∞, 0)) = 0. One can also check that µ([0, ∞)) =lim y→∞ ıy m(-ıy).

The first part of the following theorem has been shown in [START_REF] Marčenko | Distribution of eigenvalues in certain sets of random matrices[END_REF][START_REF] Silverstein | On the empirical distribution of eigenvalues of a class of large-dimensional random matrices[END_REF], and the second part in [START_REF] Bai | No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices[END_REF]: Theorem 2.1. Under Assumptions 1, 2 and 3, the following hold true:

(1) For any z ∈ C + , the equation

m = -z + t 1 + cmt ν(dt) -1 (3)
admits a unique solution m ∈ C + . The function m = m(z) so defined on C + is the Stieltjes Transform of a probability measure µ whose support is a compact set of R + . Let (λ n i ) i=1,...,N be the eigenvalues of n -1 Y n Y * n , and let

θ n = N -1 N i=1 δ λ n i
be the spectral measure of this matrix. Then for every bounded and continuous real function f ,

f (t)θ n (dt) a.s. ----→ n→∞ f (t)µ(dt). ( 4
) (2) For any interval [x 1 , x 2 ] ⊂ R -supp(µ), ♯{i : λ n i ∈ [x 1 ,
x 2 ]} = 0 with probability 1 for all large n. We now consider the additive deformation P n : Assumption 4. The deterministic N × n matrices P n have a fixed rank equal to r. Moreover, p max = sup n P n < ∞.

In order for some of the eigenvalues of Σ n Σ * n to converge to values outside supp(µ), an extra assumption involving in some sense the interaction between P n and D n is needed. Let P n = U GS n (R GS n ) * be the Gram-Schmidt factorization of P n where U GS n is an isometry N × r matrix and where (R GS n ) * is an upper triangular matrix in row echelon form whose first nonzero coefficient of each row is positive. The factorization so defined is then unique. Define the r × r Hermitian nonnegative matrix-valued measure Λ GS n as

Λ GS n (dt) = (R GS n ) *    δ d n 1 (dt) . . . δ d n n (dt)    R GS n .
Assumption 3 shows that

d max = sup n D n < ∞. Moreover, it is clear that the support of Λ GS n is included in [0, d max ] and that Λ GS n ([0, d max ]) ≤ p 2 max I r . Since the sequence Λ GS n ([0, d max ]
) is bounded in norm, for every sequence of integers increasing to infinity, there exists a subsequence n k and a nonnegative finite measure

Λ * such that f (t)Λ GS n k (dt) → f (t)Λ * (dt) for every function f ∈ C([0, d max ]), with C([0, d max ]
) being the set of continuous functions on [0, d max ]. This fact is a straightforward extension of its analogue for scalar measures. This assumption on the interaction between P n and D n appears to be the least restrictive assumption ensuring the convergence of the outliers to fixed values outside supp(µ) as n → ∞. If we consider some other factorization

P n = U n R * n of P n
where U n is an isometry matrix with size N × r, and if we associate to the R n the sequence of r × r Hermitian nonnegative matrix-valued measures Λ n defined as

Λ n (dt) = R * n    δ d n 1 (dt) . . . δ d n n (dt)    R n ( 5
)
then it is clear that Λ n (dt) = W n Λ GS n (dt)W * n for some r × r unitary matrix W n . By the compactness of the unitary group, Assumption 5 is satisfied for Λ GS n if and only if it is satisfied for Λ n . The main consequence of this assumption is that for any function f ∈ C([0, d max ]), the eigenvalues of the matrix f (t)Λ n (dt) arranged in some given order will converge.

An example taken from the fields of signal processing and wireless communications might help to have a better understanding the applicability of Assumption 5. In these fields, the matrix P n often represents a multidimensional radio signal received by an array of N antennas. Frequently this matrix can be factored as

P n = n -1/2 U n A n S * n where U n is a deterministic N × r isometry matrix, A n is a deterministic r × r matrix such that A n A * n converges to a matrix M as n → ∞ (one often assumes A n A * n = M for each n), and S n = [S ij ] n,r i,j=1
is a n × r random matrix independent of X n with iid elements satisfying ES 1,1 = 0 and E|S 1,1 | 2 = 1 (in the wireless communications terminology, U n A n is the so called MIMO channel matrix and S n is the so called signal matrix, see [38]). Taking R n = n -1/2 S n A * n in ( 5) and applying the law of large numbers, one can see that for any f ∈ C([0, d max ]), the integral f (t)Λ n (dt) converges to f (t)Λ * (dt) with Λ * (dt) = ν(dt) × M . Clearly, the accumulation points of the measures obtained from any other sequence of factorizations of P n are of the form ν(dt)×W M W * where W is an r×r unitary matrix.

It is shown in [START_REF] Silverstein | Analysis of the limiting spectral distribution of largedimensional random matrices[END_REF] that the limiting spectral measure µ has a continuous density on R * = R-{0} (see Prop. 3.1 below). Our first order result addresses the problem of the presence of isolated eigenvalues of Σ n Σ * n in any compact interval outside the support of this density. Of prime importance will be the r × r matrix functions

H * (z) = m(z) 1 + cm(z)t Λ * (dt)
where Λ * is an accumulation point of a sequence Λ n . Since The support of µ may consist in several connected components corresponding to as many "bulks" of eigenvalues. Our first theorem specifies the locations of the outliers between any two bulks and on the right of the last bulk. It also shows that there are no outliers on the left of the first bulk: ♯{i : λn i ∈ (0, A ′ ]} = 0 with probability 1 for all large n. Given any sequence of positive real numbers ρ 1 ≤ • • • ≤ ρ r lying in a connected component of supp(µ) c after the first bulk, it would be interesting to see whether there exists a sequence of matrices P n that produces outliers converging to these ρ k . The following theorem answers this question positively: Theorem 2.3. Let Assumptions 1, 2 and 3 hold true. Let ρ 1 ≤ • • • ≤ ρ r be a sequence of positive real numbers lying in a connected component (a, b) of supp(µ) c , and such that a > A. Then there exists a sequence of matrices P n satisfying Assumptions 4 and 5 such that for any compact interval I ⊂ (a, b) with {ρ 1 , . . . , ρ r } ∩ ∂I = ∅, ♯{i : λn i ∈ I} = ♯{i : ρ i ∈ I} with probability 1 for all large n. It would be interesting to complete the results of these theorems by specifying the indices of the outliers λn i that appear between the bulks. This demanding analysis might be done by following the ideas of [START_REF] Capitaine | The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations[END_REF] or [START_REF] Vallet | Improved subspace estimation for multivariate observations of high dimension: The deterministic signals case[END_REF] relative to the so called separation of the eigenvalues of Σ n Σ * n . Another approach dealing with the same kind of problem is developed in [START_REF] Bai | Exact separation of eigenvalues of large-dimensional sample covariance matrices[END_REF].

|1 + cm(z)t| = |z(1 + cm(z)t)|/|z| ≥ |ℑ(z(1 + cm(z)t))|/|z| ≥ ℑ(z)/|z| on C + , the function H * (z) is analytic on C + . It
A case of practical importance at least in the domain of signal processing is described by the following assumption: Assumption 6. The accumulation points Λ * are of the form ν(dt)×W ΩW * where

Ω =    ω 2 1 I j1 . . . ω 2 t I jt    > 0, ω 2 1 > • • • > ω 2 t , j 1 + • • • + j t = r
and where W is a unitary matrix.

Because of the specific structure of S n in the factorization P n = n -1/2 U n A n S * n , the MIMO wireless communication model described above satisfies this assumption, the ω 2 i often referring to the powers of the radio sources transmitting their signals to an array of antennas. Another case where Assumption 6 is satisfied is the case where P n is a random matrix independent of X n , where its probability distribution is invariant by right multiplication with a constant unitary matrix, and where the r non zero singular values of P n converge almost surely towards constant values.

When this assumption is satisfied, we obtain the following corollary of Theorem 2.2 which exhibits some sort of phase transition analogous to the so-called BBP phase transition [START_REF] Baik | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices[END_REF]: Depending on the value of ω 2 ℓ , ℓ = 1, . . . , t, the equation ω 2 ℓ g(x) = 1 has either zero or one solution in (a, b). Denote by ρ 1 , . . . , ρ k , k ≤ r these solutions counting multiplicities. Then the conclusion of Theorem 2.2-(1) hold true for these ρ i .

We now turn to the second order result, which will be stated in the simple and practical framework of Assumption 6. Actually, a stronger assumption is needed: Assumption 7. The following facts hold true:

sup n √ n|c n -c| < ∞, lim sup n √ n 1 t -x ν n (dt) - 1 t -x ν(dt) < ∞ for all x ∈ R -supp(ν).
Moreover, there exists a sequence of factorizations of P n such that the measures Λ n associated with these factorizations by (5) converge to ν(dt) × Ω and such that

lim sup n √ n 1 t -x Λ n (dt) - 1 t -x ν(dt) × Ω < ∞ for all x ∈ R -supp(ν)
Note that one could have considered the above superior limits to be zero, which would simplify the statement of Theorem 2.4 below. However, in practice this is usually too strong a requirement, see e.g. the wireless communications model discussed after Assumption 5 for which the fluctuations of Λ n are of order n -1/2 . On the opposite, slower fluctuations of Λ n would result in a much more intricate result for Theorem 2.4, which we do not consider here.

Before stating the second order result, a refinement of the results of Theorem 2.1-(1) is needed: [START_REF] Silverstein | On the empirical distribution of eigenvalues of a class of large-dimensional random matrices[END_REF][START_REF] Hachem | Deterministic equivalents for certain functionals of large random matrices[END_REF][START_REF] Hachem | A new approach for mutual information analysis of large dimensional multi-antenna chennels[END_REF]). Assume that D n is a n × n diagonal nonnegative matrix. Then, for any n, the equation

Proposition 2.1 ([
m n = -z 1 + 1 n Tr D n T n -1
where

T n = [-z (I n + c n m n D n )] -1
admits a unique solution m n ∈ C + for any z ∈ C + . The function m n = m n (z) so defined on C + is the Stieltjes Transform of a probability measure µ n whose support is a compact set of R + . Moreover, the n×n diagonal matrix-valued function

T n (z) = [-z(I n +c n m n (z)D n )] -1 is analytic on C + and n -1 Tr T n (z) coincides with the Stieltjes Transform of c n µ n + (1 -c n )δ 0 .
Let Assumption 2 hold true, and assume that sup n D n < ∞, and 0

< lim inf c n ≤ lim sup c n < ∞. Then the resolvents Q n (z) = (n -1 Y n Y * n -zI N ) -1 and Q n (z) = (n -1 Y * n Y n -zI n ) -1 satisfy 1 N Tr (Q n (z) -m n (z)I N ) a.s.
----→ n→∞ 0 and

1 n Tr Q n (z) -T n (z) a.s. ----→ n→∞ 0 (6)
for any z ∈ C + . When in addition Assumptions 1 and 3 hold true, m n (z) converges to m(z) provided in the statement of Theorem 2.1 uniformly on the compact subsets of C + .

The function

m n (z) = -z + t(1 + c n m n (z)t) -1 ν n (dt) -1 is a finite n approx- imation of m(z). Notice that since N -1 Tr Q n (z) is the Stieltjes Transform of the spectral measure θ n of n -1 Y n Y * n , Convergence (4
) stems from [START_REF] Baik | Eigenvalues of large sample covariance matrices of spiked population models[END_REF]. We shall also need a finite n approximation of H * (z) defined as

H n (z) = m n (z) 1 + c n m n (z)t Λ n (dt).
With these definitions, we have the following preliminary proposition:

Proposition 2.2. Let Assumptions 1, 3-7 hold true. Let g be the function defined in the statement of Corollary 2.1 and let B µ = sup(supp(µ)). Assume that the equation ω 2 1 g(x) = 1 has a solution in (B µ , ∞), and denote ρ 1 > • • • > ρ p the existing solutions (with respective multiplicities j 1 , . . . , j p ) of the equations ω 2 ℓ g(x) = 1 in (B µ , ∞). Then the following facts hold true:

• ∆(ρ i ) = 1 -c m(ρ i )t 1 + cm(ρ i )t 2 ν(dt) is positive for every i = 1, . . . , p. • Denoting by H 1,n (z), . . . , H p,n (z) the first p upper left diagonal blocks of H n (z), where H i,n (z) ∈ C ji×ji , lim sup n √ n(H i,n (ρ i ) + I ji ) < ∞ for every i = 1, . . . , p.
We recall that a GUE matrix (i.e., a matrix taken from the Gaussian Unitary Ensemble) is a random Hermitian matrix G such that G ii ∼ N (0, 1), ℜ(G ij ) ∼ N (0, 1/2) and ℑ(G ij ) ∼ N (0, 1/2) for i < j, and such that all these random variables are independent. Our second order result is provided by the following theorem:

Theorem 2.4. Let Assumptions 1-7 hold true. Keeping the notations of Proposition 2.2, let

M n i = √ n       λn j1+•••+ji-1+1 . . . λn j1+•••+ji    -ρ i    1 . . . 1      
where j 0 = 0 and where the eigenvalues λn i of Σ n Σ * n are arranged in decreasing order. Let G 1 , . . . , G p be independent GUE matrices such that G i is a j i ×j i matrix. Then, for any bounded continuous f : R j1+...+jp → R,

E f M n 1 , . . . , M n p -E f Ξ n 1 , . . . , Ξ n p -→ n→∞ 0
where Ξ n i ∈ R ji is the random vector of the decreasingly ordered eigenvalues of the matrix

1 ω 2 i g(ρ i ) ′ α i G i + √ n (H i,n (ρ i ) + I ji ) ,
where

α 2 i = m 2 (ρ i ) ∆(ρ i ) t 2 + 2ω 2 i t (1 + cm(ρ i )t) 2 ν(dt) + c ω 2 i m(ρ i )t (1 + cm(ρ i )t) 2 ν(dt) 2 .
Some remarks can be useful at this stage. The first remark concerns Assumption 7, which is in some sense analogous to [START_REF] Benaych-Georges | Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices[END_REF]Hypothesis 3.1]. This assumption is mainly needed to show that the √ n H i,n (ρ i ) + I ji are bounded, guaranteeing the tightness of the vectors M n i . Assuming that Λ n and Λ ′ n both satisfy the third item of Assumption 7, denoting respectively by H i,n (ρ i ) and H ′ i,n (ρ i ) the matrices associated to these measures as in the statement of Theorem 2.4, it is possible to show that

√ n(σ(H i,n (ρ i ) -σ(H ′ i,n (ρ i )) → 0 as n → ∞.
Thus the results of this theorem do not depend on the particular measure Λ n satisfying Assumption 7. Finally, we note that Assumption 7 can be lightened at the expense of replacing the limit values ρ i with certain finite n approximations of the outliers, as is done in the applicative paper [START_REF] Vinogradova | Statistical inference in large antenna arrays under unknown noise pattern[END_REF].

The second remark pertains to the Gaussian assumption on the elements of X n . We shall see below that the results of Theorems 2.2-2.4 are intimately related to the first and second order behaviors of bilinear forms of the type

u * n Q n (x)v n , ũ * n Q n (x)ṽ n , and n -1/2 u * n Y n Q n (x)
ṽ n where u n , v n , ũn and ṽn are deterministic vectors of bounded norm and of appropriate dimensions, and where x is a real number lying outside the support of µ. In fact, it is possible to generalize Theorems 2.2 and 2.3 to the case where the elements of X n are not necessarily Gaussian. This can be made possible by using the technique of [START_REF] Hachem | On bilinear forms based on the resolvent of large random matrices[END_REF] to analyze the first order behavior of these bilinear forms. On the other hand, the Gaussian assumption plays a central role in Theorem 2.4. Indeed, the proof of this theorem is based on the fact that these bilinear forms asymptotically fluctuate like Gaussian random variables when centered and scaled by √ n. Take u n = e 1,N and ṽn = e 1,n where e k,m is the k th canonical vector of R m . We show below (see Proposition 2.1 and Lemmas 4.3 and 4.6) that the elements qn ij of the resolvent Q n (x) are close for large n to the elements tn ij of the deterministic matrix T n (x). We therefore write informally

e * 1,N Y n Q(x)e 1,n = n j=1 (d n j ) 1/2 qn j1 X 1j ≈ (d n 1 ) 1/2 tn 11 X 11 + n j=2 (d n j ) 1/2 qn j1 X 1j .
It can be shown furthermore that tn 11 = O(1) for large n and that the sum n j=2 is tight. Hence, e * 1,N Y n Q(x)e 1,n is tight. However, when X 11 is not Gaussian, we infer that e * 1,N Y n Q(x)e 1,n does not converge in general towards a Gaussian random variable. In this case, if we choose P n = ω 2 e 1,N e * 1,n (see Section 5), Theorem 2.4 no longer holds. Yet, we conjecture that an analogue of this theorem can be recovered when e 1,N and e 1,n are replaced with delocalized vectors, following the terminology of [START_REF] Capitaine | Central limit theorems for eigenvalues of deformations of Wigner matrices[END_REF]. In a word, the elements of these vectors are "spread enough" so that the Gaussian fluctuations are recovered.

A word about the notations. In the remainder of the paper, we shall often drop the subscript or the superscript n when there is no ambiguity. A constant bound that may change from an inequality to another but which is independent of n will always be denoted K.

Element (i, j) of matrix M is denoted M ij or [M ] ij . Element i of vector x is denoted [x] i .
Convergences in the almost sure sense, in probability and in distribution will be respectively denoted a.s.

-→, P -→, and D -→.

Preliminaries and useful results

We start this section by providing the main ideas of the proofs of Theorems 2.2 and 2.3.

3.1.

Proof principles of the first order results. The proof of Theorem 2.2-(1), to begin with, is based on the idea of [START_REF] Benaych-Georges | The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices[END_REF][START_REF] Benaych-Georges | The singular values and vectors of low rank perturbations of large rectangular random matrices[END_REF]. We start with a purely algebraic result.

Let P n = U n R * n be a factorization of P n where U n is a N × r isometry matrix. Assume that x > 0 is not an eigenvalue of n -1 Y n Y * n . Then x is an eigenvalue of Σ n Σ *
n if and only if det S n (x) = 0 where S n (x) is the 2r × 2r matrix

S n (x) = √ xU * n Q n (x)U n I r + n -1/2 U * n Y n Q n (x)R n I r + n -1/2 R * n Q n (x)Y * n U n √ xR * n Q n (x)
R n (for details, see the derivations in [START_REF] Benaych-Georges | The singular values and vectors of low rank perturbations of large rectangular random matrices[END_REF] or in [START_REF] Hachem | A subspace estimator for fixed rank perturbations of large random matrices[END_REF]Section 3]). The idea is now the following. Set x in supp(µ) c . Using an integration by parts formula for functionals of Gaussian vectors and the Poincaré-Nash inequality [START_REF] Pastur | Eigenvalue distribution of large random matrices[END_REF], we show that when n is large,

U * n Q n (x)U n ≈ m n (x)I r , R * n Q n (x)R n ≈ R * n T n (x)R n , and n -1/2 R * n Q n (x)Y * n U n ≈
0 by controlling the moments of the elements of the left hand members. To be able to do these controls, we make use of a certain regularizing function which controls the escape of the eigenvalues of n -1 Y n Y * n out of supp(µ). Thanks to these results, S n (x) is close for large n to

S n (x) = √ xm n (x)I r I r I r √ xR * n T n (x)R n .
Hence, we expect the eigenvalues of Σ n Σ * n in the interval I, when they exist, to be close for large n to the zeros in I of the function

det S n (x) = det xm n (x)R * n T n (x)R n -I r = (-1) r det m n (x)R * n (I n + c n m n (x)D n ) -1 R * n + I r = (-1) r det (H n (x) + I r )
which are close to the zeros of D(x) = det(H * (x) + I r ). By Assumption 5, these zeros are independent of the choice of the accumulation point Λ * .

To prove Theorems 2.2-(2) and 2.3, we make use of the results of [START_REF] Silverstein | Analysis of the limiting spectral distribution of largedimensional random matrices[END_REF] and [START_REF] Mestre | Improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates[END_REF][START_REF] Mestre | On the asymptotic behavior of the sample estimates of eigenvalues and eigenvectors of covariance matrices[END_REF] relative to the properties of µ and to those of the restriction of m(z) to R-supp(µ). The main idea is to show that A) (these x lie at the left of the first bulk) and for all t ∈ supp(ν). • For any component (a, b) ⊂ supp(µ) c such that a > A (i.e., lying between two bulks or at the right of the last bulk), there exists a Borel set E ⊂ R + such that ν(E) > 0 and

• m(x)(1 + cm(x)t) -1 > 0 for all x ∈ supp(µ) c ∩ (-∞,
q(x) = E m(x) 1 + cm(x)t ν(dt) < 0
for all x ∈ (a, b). Thanks to the first result, for any x lying if possible between zero and the left edge of the first bulk, D(x) > 0, hence Σ n Σ * n has asymptotically no outlier at the left of the first bulk. Coming to Theorem 2.3, let E be a set associated to (a, b) by the result above. We build a sequence of matrices P n of rank r, and such that the associated Λ n have an accumulation point of the form Λ * (dt) = ½ E (t) Ω ν(dt) where we choose Ω = diag(-q(ρ 1 ) -1 , . . . , -q(ρ r ) -1 ). Theorem 2.2- [START_REF] Bai | Central limit theorems for eigenvalues in a spiked population model[END_REF] shows that the function H * (x) = q(x)Ω associated with this Λ * is increasing on (a, b). As a result, H * (x)+I r becomes singular precisely at the points ρ 1 , . . . , ρ r .

3.2. Sketch of the proof of the second order result. The fluctuations of the outliers will be deduced from the fluctuations of the elements of the matrices S n (ρ i ) introduced above. The proof of Theorem 2.4 can be divided into two main steps. The first step (Lemma 5.4) consists in establishing a Central Limit Theorem on the 3p-uple of random matrices

√ n U * i,n Y n Q n (ρ i )R i,n √ n , U * i,n (Q n (ρ i ) -m n (ρ i )I N )U i,n , R * i,n ( Q n (ρ i ) -T n (ρ i ))R i,n p i=1
, where P n = U n R * n is a sequence of factorization such that Λ n satisfies the third item of Assumption 7. We also write

U n = [U 1,n • • • U t,n ] and R n = [R 1,n • • • R t,n ] where U i,n ∈ C N ×ji and R i,n ∈ C n×ji .
This CLT is proven by using the Gaussian tools introduced in Section 3.4, namely the integration by parts formula and the Poincaré-Nash inequality, and by relying on the invariance properties of the probability distribution of n -1/2 X n . The fluctuations of the zeros of det S n (x) outside supp(µ) are then deduced from this CLT by adapting the approach of [START_REF] Benaych-Georges | Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices[END_REF] (Lemma 5.5).

We now come to the basic mathematical tools needed for our proofs:

X = ℓ λ ℓ v ℓ v * ℓ , let adj(X) = k ( ℓ =k λ ℓ )v k v *
k be the classical adjoint of X, i.e., the transpose of its cofactor matrix. Let ψ be a continuously differentiable real-valued function on R. Then

∂ det ψ n -1 Y Y * ∂ Ȳij = 1 n adj ψ n -1 Y Y * ψ ′ n -1 Y Y * y j i
where y j is column j of Y , see [19, Lemma 3.9] for a proof. We shall also need the expressions of the following derivatives of the elements of the resolvents Q and Q (see [START_REF] Hachem | A new approach for mutual information analysis of large dimensional multi-antenna chennels[END_REF]):

∂Q pq ∂ Ȳij = - 1 n [QY ] pj Q iq , ∂ Q pq ∂ Ȳij = - 1 n Q pj [Y Q] iq .
3.4. Gaussian tools. Our analysis fundamentally relies on two mathematical tools which are often used in the analysis of large random matrices with Gaussian elements. The first is the so called integration by parts (IP) formula for functionals of Gaussian vectors introduced in random matrix theory in [START_REF] Khorunzhy | Limits of infinite interaction radius, dimensionality and the number of components for random operators with off-diagonal randomness[END_REF][START_REF] Pastur | On an asymptotic property of the spectrum of the sum of one-dimensional independent random operators[END_REF]. Let Γ : R 2N n → C be a continuously differentiable function polynomially bounded together with its partial derivatives. Then

E (Y ij Γ(Y )) = d j E ∂Γ(Y ) ∂ Ȳij
for any i ∈ {1, . . . , N } and j ∈ {1, . . . , n}. The second tool is the Poincaré-Nash inequality (see for instance [START_REF] Chen | An inequality for the multivariate normal distribution[END_REF]). In our situation, it states that the variance Var(Γ(Y )) satisfies

Var(Γ(Y )) ≤ N i=1 n j=1 d j E ∂Γ(Y ) ∂Y ij 2 + ∂Γ(Y ) ∂ Ȳij 2 .
We now recall the results of Silverstein and Choi [START_REF] Silverstein | Analysis of the limiting spectral distribution of largedimensional random matrices[END_REF] which will be needed to prove Theorems 2.2-(2) and 2.3. Close results can be found in [START_REF] Mestre | Improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates[END_REF] and in [START_REF] Mestre | On the asymptotic behavior of the sample estimates of eigenvalues and eigenvectors of covariance matrices[END_REF]. In [START_REF] Silverstein | Analysis of the limiting spectral distribution of largedimensional random matrices[END_REF], the support of µ is also identified. Since m(z) is the unique solution in C + of (3) for z ∈ C + , it has a unique inverse on C + given by

z(m) = - 1 m + t 1 + cmt ν(dt)
The characterization of the support of µ is based on the following idea. On any open interval of supp(µ) c , m(x) = (tx) -1 µ(dt) is a real, continuous and increasing function. Consequently, it has a real, continuous and increasing inverse. In [START_REF] Silverstein | Analysis of the limiting spectral distribution of largedimensional random matrices[END_REF], it is shown that the converse is also true. More precisely, let B = {m : m = 0, -(cm) -1 ∈ supp(ν) c }, and let

x : B -→ R m -→ x(m) = - 1 m + t 1 + cmt ν(dt). (7)
Then the following proposition holds: The following proposition will also be useful:

Proposition 3.3 ([37], Th. 4.4). Let [m 1 , m 2 ] and [m 3 , m 4 ] be two disjoint in- tervals of B satisfying ∀m ∈ (m 1 , m 2 ) ∪ (m 3 , m 4 ), x ′ (m) > 0. Then [x 1 , x 2 ] and [x 3 , x 4 ] are disjoint where x i = x(m i ).
The following result is also proven in [START_REF] Silverstein | Analysis of the limiting spectral distribution of largedimensional random matrices[END_REF]:

Proposition 3.4. Assume that ν({0}) = 0. Then µ({0}) = max(0, 1 -c -1 ).
We shall assume hereafter that ν({0}) = 0 without loss of generality (otherwise, it would be enough to change the value of c). The two following lemmas will also be needed: Lemma 3.1. Let I be a compact interval of supp(µ) c , and let D I be the closed disk having I as one of its diameters. Then there exists a constant K which depends on I only such that

∀ t ∈ supp(ν), ∀ z ∈ D I , |1 + cm(z)t| ≥ K, and ∀ n large enough, ∀ t ∈ supp(ν n ), ∀ z ∈ D I , |1 + c n m n (z)t| ≥ K.
From the second inequality, we deduce that if {0} ∈ I, the matrix function T n (z) is analytic in a neighborhood of I for n large enough, and lim sup

n sup z∈DI T n (z) < ∞. (8)
Proof. When z ∈ C + , ℑm(z) > 0 and ℑ(-(cm(z)) -1 ) > 0, and we have the opposite inequalities when ℑz < 0. Applying Proposition 3. Let

f n (z) = d(-(c n m n (z)) -1 , supp(ν n )).
We have

f n (z) ≤ d -1 c n m n (z) , -1 cm(z) + d -1 cm(z) , supp(ν n ) ≤ d -1 c n m n (z) , -1 cm(z) + f (z) + d H (supp(ν n ), supp(ν)),
and Indeed, one can easily check the conditions that enable ũ * n T n (z)ũ n to be a Stieltjes Transform of a positive finite measure. The last result is obtained by analyticity in a neighborhood of I. In fact, it can be checked that supp(γ n ) ⊂ supp(µ n ) ∪ {0}.

f (z) ≤ d(-(c n m n (z)) -1 , -(cm(z)) -1 ) + f n (z) + d H (supp(ν n ), supp (

3.6.

A control over the support of θ n . In this paragraph, we adapt to our case an idea developed in [START_REF] Capitaine | The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations[END_REF] to deal with Wigner matrices whose elements distribution satisfies a Poincaré-Nash inequality. Proposition 3.5. For any sequence of n × n deterministic diagonal nonnegative matrices U n such that sup n U n < ∞,

1 n Tr EQ n (z) -m n (z) ≤ P (|z|)R(|ℑ(z)| -1 ) n 2
, and

1 n Tr U n E Q n (z) - 1 n Tr U n T n (z) ≤ P (|z|)R(|ℑ(z)| -1 ) n 2
for z ∈ C + , where P and R are polynomials with nonnegative coefficients independent of n.

This proposition is obtained from a simple extension of the results of [18, Th. Since N -1 Tr Q n (z) is the Stieltjes Transform of the spectral measure θ n , the inversion formula [START_REF] Bai | Methodologies in spectral analysis of large-dimensional random matrices, a review[END_REF] shows that

ϕ(t) θ n (dt) = 1 π lim y↓0 ℑ ϕ(x) 1 N Tr Q n (x + ıy) dx for any function ϕ ∈ C ∞ c (R, R).
Using then Proposition 3.5 and Lemma 3.3, we obtain the following result:

Proposition 3.6. For any function ϕ ∈ C ∞ c (R, R), E ϕ(t) θ n (dt) - ϕ(t) µ n (dt) ≤ K n 2 .

Proofs of first order results

In all this section, I is a compact interval of a component (a, b) of supp(µ) c , and z is a complex number such that ℜ(z) ∈ I and ℑ(z) is arbitrary. Moreover, u n , v n ∈ C N and ũn , ṽn ∈ C n are sequences of deterministic vectors such that sup n max( u n , v n , ũn , ṽn ) < ∞, and U n is a sequence of n × n diagonal deterministic matrix such that sup n U n < ∞. We now introduce the regularization function alluded to in the introduction. Choose ε > 0 small enough so that I ∩ S ε = ∅ where S ε = {x ∈ R, d(x, supp(µ) ∪ {0}) ≤ ε}. Fix 0 < ε ′ < ε, let ψ : R → [0, 1] be a continuously differentiable function such that

ψ(x) = 1 if x ∈ S ε ′ 0 if x ∈ R -S ε and let φ n = det ψ(n -1 Y n Y * n ).
In all the subsequent derivations, quantities such as u * n Q n (z)u n or ũ * n Q n (z)ũ n for ℜ(z) ∈ I will be multiplied by φ n in order to control their magnitudes when z is close to the real axis. By performing this regularization as is done in [START_REF] Hachem | Large information plus noise random matrix models and consistent subspace estimation in large sensor networks[END_REF], we shall be able to define and control the moments of random variables such as

φ n u * n Q n (z)u n or φ n ũ * n Q n (z)
ũ n with the help of the Gaussian tools introduced in Section 3.4.

We start with a series of lemmas. The first of these lemmas relies on Proposition 3.6 and on the Poincaré-Nash inequality. Its detailed proof is a minor modification of the proof of [19, Lemma 3] and is therefore omitted: Lemma 4.1. Given 0 < ε ′ < ε, let ϕ be a smooth nonnegative function equal to zero on S ε ′ and to one on R -S ε . Then for any ℓ ∈ N, there exists a constant K ℓ for which

E Tr ϕ(n -1 Y n Y * n ) ℓ ≤ K ℓ n ℓ . Remark 1.
Notice that this lemma proves Theorem 2.1-(2). The proof provided in [START_REF] Bai | No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices[END_REF] is in fact more general, being not restricted to the Gaussian case. Lemma 4.2. For any ℓ ∈ N, the following holds true:

E      N,n i,j=1 d j ∂φ n ∂ Ȳij 2   ℓ    ≤ K ℓ n 2ℓ . Proof. Letting n -1/2 Y = W diag( √ λ 1 , • • • , √ λ N )V * be a singular value decompo- sition of n -1/2 Y , we have adj ψ Y Y * n ψ ′ Y Y * n Y √ n = W ΞV * where Ξ = diag √ λ k ψ ′ (λ k ) ℓ =k ψ(λ ℓ ) N k=1
and we observe that Tr Ξ 2 ≤ KZ n where Z n = ♯{k :

λ k ∈ S ε -S ε ′ }.
Using the first identity in Section 3.3 and recalling that | Tr(AB)| ≤ A Tr B when A is a square matrix and B is a Hermitian nonnegative matrix, we have

E      N,n i,j=1 d j ∂φ n ∂ Ȳij 2   ℓ    = 1 n ℓ E Tr adj(ψ)ψ ′ Y DY * n adj(ψ)ψ ′ ℓ ≤ K n ℓ EZ ℓ n
and the result follows from Lemma 4.1 with a proper choice of ϕ.

Lemma 4.3. The following inequalities hold true:

E |φ n u * n Q n (z)v n -E[φ n u * n Q n (z)v n ]| 4 ≤ K n 2 , E φ n ũ * n Q n (z)ṽ n -E[φ n ũ * n Q n (z)ṽ n ] 4 ≤ K n 2 , Var (φ n Tr Q n (z)) ≤ K.
Proof. We shall only prove the first inequality. By the polarization identity, this inequality is shown whenever we show that

E |φ u * Qu -E[φ u * Qu]| 4 ≤ K/n 2 .
Let us start by showing that Var(φ u * Qu) ≤ K/n. By the Poincaré-Nash inequality, we have

Var (φu * Qu) ≤ 2 N,n i,j=1 d j E ∂φu * Qu ∂ Ȳij 2 ≤ 4 N,n i,j=1 d j E φ ∂u * Qu ∂ Ȳij 2 + 4 N,n i,j=1 d j E u * Qu ∂φ ∂ Ȳij 2 .
Using the expression of ∂Q pq /∂ Ȳij in Section 3.3, we have

∂u * Qu/∂ Ȳij = -n -1 u * Qy j [Qu] i , hence N,n i,j=1 d j E φ ∂u * Qu ∂ Ȳij 2 = 1 n E φ 2 u * Q Y DY * n Qu u * Q 2 u ≤ K n
since the argument of the expectation is bounded for ℜ(z) ∈ I. From the first identity in Section 3.

X= X -EX, E • φ u * Qu 4 = (Var(φ u * Qu)) 2 + Var • φ u * Qu 2 ≤ K/n 2 + Var • φ u * Qu 2 .
By the Poincaré-Nash inequality,

Var • φ u * Qu 2 ≤ 2 N,n i,j=1 d j E ∂ • φ u * Qu 2 / ∂ Ȳij 2 ≤ 16 N,n i,j=1 d j E • φ u * Qu φ ∂u * Qu ∂ Ȳij 2 + 16 N,n i,j=1 d j E • φ u * Qu u * Qu ∂φ ∂ Ȳij 2 := V 1 + V 2 .
By developing the derivative in

V 1 similarly to above, V 1 ≤ Kn -1 E • φ u * Qu 2 ≤ Kn -2
. By the Cauchy-Schwarz inequality and Lemma 4.2,

V 2 ≤ K N,n i,j=1 d j E • φ u * Qu ∂φ ∂ Ȳij 2 ≤ K n 2 E • φ u * Qu 4 1/2 . Writing a n = n 2 E • φ u * Qu 4 , we have shown that √ a n ≤ K/ √ a n + K/n.
Assume that a n is not bounded. Then there exists a sequence n k of integers such that a n k → ∞, which raises a contradiction. The first inequality in the statement of this lemma is shown. The other two inequalities can be shown similarly.

Lemma 4.4. The following holds true:

1 -Eφ n ≤ K ℓ n ℓ for any ℓ ∈ N.
Proof. For 0 < ε 1 < ε ′ where ε ′ is defined in the construction of ψ, let ϕ be a smooth nonnegative function equal to zero on S ε1 and to one on R -S ε ′ . Then 1φ n ≤ Tr(ϕ(n -1 Y Y * )) ℓ for any ℓ ∈ N, and the result stems from Lemma 4.1.

Lemma 4.5. The following inequalities hold true (recall that ℜ(z) ∈ supp(µ)):

|E [φ n Tr Q n (z)] -N m n (z)| ≤ K n , and
Tr U n E[φ n Q n (z)] -T n (z) ≤ K n .
Proof. Let ε be defined in the construction of ψ. Choose a small ε 1 > ε in such a way that S ε1 ∩ I = ∅. Let ζ be a C ∞ c (R, R) nonnegative function equal to one on S ε and to zero on R -S ε1 , so that

φ 1 n Tr Q = φ ζ(t) t -z θ n (dt) .
Using this equality, and recalling that φ ∈ [0, 1], we have

Eφ 1 n Tr Q -E ζ(t) t -z θ n (dt) ≤ E (1 -φ) ζ(t) t -z θ n (dt) ≤ 1 -Eφ d(z, S ε1 ) ≤ K ℓ n ℓ
for any ℓ ∈ N. Moreover, we have

E ζ(t) t -z θ n (dt) -m n (z) = E ζ(t) t -z θ n (dt) - ζ (
|E[φ n u * n Q n (z)v n ] -u * n v n m n (z)| ≤ K/n, E[φ n ũ * n Q n (z)ṽ n ] -ũ * n T n (z)ṽ n ≤ K/n.
In [START_REF] Hachem | On bilinear forms based on the resolvent of large random matrices[END_REF], it is proven in a more general setting that

Eu * n Q n (z)u n -u n 2 m n (z) ≤ P (|z|)R(|ℑ(z)| -1 )/ √ n for any z ∈ C + . Observing that u * n Q n (z)u n and u n 2 m n (z)
are Stieltjes Transforms of positive measures, and mimicking the proof of the previous lemma, we can establish this lemma with the rate O(n -1/2 ), which is in fact enough for our purposes. However, in order to give a flavor of the derivations that will be carried out in the next section, we consider here another proof that uses the IP formula and the Poincaré-Nash inequality. To that end, we introduce new notations:

β(z) = φ n 1 n Tr Q n (z), α(z) = Eβ(z), β(z) = β(z) -φ n α(z), and
α(z) = 1 n Tr D n [-z(I n + α(z)D n )] -1 .
Proof. We start with the first inequality. By the IP formula, we have

E[Q pi Y ij Ȳℓj φ] = - d j n E[[Qy j ] p Q ii Ȳℓj φ]+δ(ℓ-i)d j E[Q pi φ]+ d j n E[Q pi Ȳℓj [adj(ψ)ψ ′ y j ] i ].
Taking the sum over i, we obtain

E[[Qy j ] p Ȳℓj φ] = -d j E[[Qy j ] p Ȳℓj β] + d j E[Q pℓ φ] + d j n E[ Ȳℓj [Q adj(ψ)ψ ′ y j ] p ]. Writing β = β + φα, we get E[[Qy j ] p Ȳℓj φ] = d j 1 + αd j E[Q pℓ φ] - d j 1 + αd j E[[Qy j ] p Ȳℓj β] + d j n(1 + αd j ) E[ Ȳℓj [Q adj(ψ)ψ ′ y j ] p ].
the sum over j, we obtain

E Q Y Y * n pℓ φ = -z αE[Q pℓ φ] -E β Q Y D(I + αD) -1 Y * n pℓ + 1 n E Q adj(ψ)ψ ′ Y D(I + αD) -1 Y * n pℓ .
We now use the identity zQ = n -1 QY Y * -I, which results in

zE[Q pℓ φ] = E Q Y Y * n pℓ φ -δ(p -ℓ)E[φ], E[Q pℓ φ] = δ(p -ℓ) -z(1 + α) E[φ] +
2nd and 3rd terms of next to last equation z(1 + α) .

Multiplying each side by [u * ] p [v] ℓ and taking the sum over p and ℓ, we finally obtain

E[u * Qvφ] = E[φ] u * v -z(1 + α) -[-z(1 + α)] -1 E βu * Q Y D(I + αD) -1 Y * n v + 1 n [-z(1 + α)] -1 E u * Q adj(ψ)ψ ′ Y D(I + αD) -1 Y * n v . (9)
Let us evaluate the three terms at the right hand side of this equality. From The second result in the statement of the lemma is proven similarly. The proof requires the second inequality of Lemma 4.5.

The proof of the following lemma can be done along the same lines and will be omitted:

Lemma 4.7. The following inequalities hold true:

Eφ n u * n Y n Q n (z)ṽ n ≤ K/ √ n E φ n u * n Y n Q n (z)ṽ n 4 ≤ K.
We now prove Theorem 2. 1), formalizing the argument introduced in Section 3.1. In the remainder, we restrict ourselves to the probability one set where n -1 Y Y * has no eigenvalues for large n in a large enough closed interval in (a, b). Given a compact interval I ⊂ (a, b), let D • I be the open disk with diameter I. Observe that the functions S n (x) and S n (x) introduced in Section 3.1 can be extended analytically to a neighborhood of D I , and the determinants of these functions do not cancel outside the real axis. Let L n = ♯{i : λn

i ∈ D • I }, L n = ♯{ zeros of det S n (z) in D • I }, and L = ♯{i : ρ i ∈ D • I }.
We need to prove that L n = L with probability one for n large. By the argument of [START_REF] Benaych-Georges | The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices[END_REF][START_REF] Benaych-Georges | The singular values and vectors of low rank perturbations of large rectangular random matrices[END_REF], L n is equal to the number of zeros of det S n (z) in I • . By the well known argument principle for holomorphic functions,

L n = 1 2ıπ ∂D • I (det S n (z)) ′ det S n (z) dz, L n = 1 2ıπ ∂D • I (det S n (z)) ′ det S n (z) dz = 1 2ıπ ∂D • I (det(H n (z) + I r )) ′ det(H n (z) + I r ) dz and L = 1 2ıπ ∂D • I (det(H * (z) + I r )) ′ det(H * (z) + I r ) dz
where ∂D • I is seen as a positively oriented contour.

For any 1 ≤ k, ℓ ≤ r, let h n,k,ℓ (z) = [U * n (Q n (z) -m n (z)I N )U n ] k,ℓ .
Let V be a small neighborhood of D I , the closure of D • I . Let z m be a sequence of complex numbers in V having an accumulation point in V . By Lemmas 4.1, 4.3 and 4.6 and the Borel Cantelli lemma, h n,k,ℓ (z m ) as → 0 as n → ∞ for every m. Moreover, for n large, the h n,k,ℓ are uniformly bounded on any compact subset of V . By the normal family theorem, every n-sequence of h n,k,ℓ contains a further subsequence which converges uniformly on the compact subsets of V to a holomorphic function h * . Since h * (z m ) = 0 for every m, we obtain that almost surely, h n,k,ℓ converges uniformly to zero on the compact subsets of V , and the same can be said about

U * n (Q n (z) -m n (z)I N )U n .
Using in addition Lemmas 3.1 and 4.7 we obtain the same result for

R * n ( Q n (z) -T n (z))R n and n -1/2 U * n Y n Q n (z)R n .
Since det X is a polynomial in the elements of matrix X, det S n (z)det S n (z) converges almost surely to zero on ∂D • I , and this convergence is uniform. By analyticity, the same can be said about the derivatives of these quantities. Moreover, det S n (z) converges to (-1) r det(H * (z) + I r ) (which is the same for all accumulation points Λ * ) uniformly on ∂D • I . Similarly, (det S n (z)) ′ converges to (-1) r (det(H * (z) + I r )) ′ uniformly on ∂D • I . Furthermore, by construction of the interval I, we have inf z∈∂D

(i) If b ≤ A, then m(x)(1+cm(x)t) -1 > 0 for all x ∈ (a, b) and all t ∈ supp(ν).
Alternatively, if a > A, then there exists a Borel set E ⊂ R + such that ν(∂E) = 0 and

q(x) = E m(x) 1 + cm(x)t ν(dt) < 0 for all x ∈ (a, b).
Proof. The proof is based on the results of Section 3.5. To have an illustration of some of the proof arguments, the reader may refer to Figures 1 and2 which provide typical plots of x(m) for c < 1 and c > 1 respectively. We start by fixing a point x 0 in (a, b), we write m 0 = m(x 0 ) and we choose in the remainder of the proof E = [0, -(cm 0 ) -1 ] with the convention E = ∅ when m 0 > 0.

We already assumed that ν({0}) = 0 in Section 3.5. Since -(cm 0 ) -1 ∈ supp(ν) c by Proposition 3.2, ν({-(cm 0 ) -1 }) = 0. Hence ν(∂E) = 0. To prove the lemma, we shall show that

ν(E) > 0 ⇔ a > A. ( 10
)
To see why [START_REF] Bolotnikov | On a general moment problem on the half axis[END_REF] proves the lemma, consider first a > A. Then m 0 < 0 since ν(E) > 0. Moreover 1+cm 0 t ≥ 0 for any t ∈ E. It results that q(x 0 ) < 0. Consider now another point x 1 ∈ (a, b), and let m 1 = m(x 1 ) and

E 1 = [0, -(cm 1 ) -1 ]
. By the same argument as for x 0 , we get that E1 m(x)(1 + cm(x)t) -1 ν(dt) < 0. But Proposition 3.2 shows that the closed interval between m 0 and m 1 belongs to the set B. It results that ν(E △ E 1 ) = 0 where E △ E 1 is the symmetric difference between E and E 1 . Hence q(x 1 ) < 0 and (ii) is true. Assume now that b ≤ A. If m 0 > 0, then for all x 1 ∈ (a, b), m 1 = m(x 1 ) > 0 since the segment between m 0 and m 1 belongs to B, and (i) is true. Assume that m 0 < 0. Then since ν(E) = 0, for any t ∈ supp(ν), t > -(cm 0 ) -1 , hence m 0 (1 + cm 0 t) -1 > 0. If we take another point x 1 ∈ (a, b), then the associated set E 1 will also satisfy ν(E 1 ) = 0 since ν(E △ E 1 ) = 0. Hence we also have m 1 (1 + cm 1 t) -1 > 0 for any t ∈ supp(ν), which proves (i).

We now prove [START_REF] Bolotnikov | On a general moment problem on the half axis[END_REF] in the ⇐ direction, showing that x 0 > A ⇒ ν(E) > 0. Since m(z) is the Stieltjes Transform of a probability measure supported by R + , the function m(x) decreases to zero as x → -∞. Furthermore, (0, ∞) belongs to B. Hence, by Proposition 3.2, x 0 > A ⇒ m 0 < 0. Assume that ν(E) = 0. Then (-∞, m 0 ] ⊂ B. Since t > -(cm 0 ) -1 in the integral in [START_REF] Benaych-Georges | Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices[END_REF], x(m) → 0 as m → -∞ by the dominated convergence theorem. By Propositions 3.1, 3.2 and 3.3, x(m) should be increasing from 0 to x 0 on (-∞, m 0 ]. This contradicts x 0 > A.

We now prove [START_REF] Bolotnikov | On a general moment problem on the half axis[END_REF] in the ⇒ direction. To that end, we consider in turn the cases c < 1, c > 1 and c = 1. Assume c < 1. Since m(z) is the Stieltjes Transform of a probability measure supported by R + , the function m(x) decreases to zero as x → -∞. Hence x(m) → -∞ as m → 0 + . From [START_REF] Benaych-Georges | Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices[END_REF] we notice that mx(m) → (1c)/c > 0 as m → ∞, hence x(m) reaches a positive maximum on (0, ∞). By Propositions 3.2 and 3.3, this maximum is A, and we have x < A ⇒ m(x) > 0. Therefore, x 0 < A ⇒ ν(E) = 0. Consider now the case c > 1. We shall also show that x 0 < A ⇒ ν(E) = 0. By Proposition 3.4, the measure µ has a Dirac at zero with weight 1c -1 . Hence, either x 0 < 0, or A > 0 and 0 < x 0 < A. Since m(z) is the Stieltjes Transform of a probability measure supported by R + , it holds that x < 0 ⇒ m(x) > 0. Hence, ν(E) = 0 when x 0 < 0. We now consider the second case. Since (0, -∞. Hence this image is (-∞, m 0 ]. This implies that ν(E) = 0. We finally consider the case c = 1. We show here that A = 0, which will result in

x 0 < A ⇒ m 0 > 0 ⇒ ν(E) = 0 as above. Assume A > 0 and let x 1 ∈ (0, A). By Proposition 3.4, µ({0}) = 0 hence m(x 1 ) = (t -x 1 ) -1 µ(dt) > 0.
But from [START_REF] Benaych-Georges | Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices[END_REF], we easily show that x(m) increases from -∞ to 0 as m increases from 0 to ∞, which raises a contradiction. This concludes the proof of Lemma 4.8.

This lemma shows that for any x < inf(supp(µ) -{0}), H * (x) ≥ 0, hence D(x) > 0 for those x. This proves Theorem 2.2-(2). Turning to Theorem 2.3, let E be a Borel set associated to (a, b) by Lemma 4.8-(ii) and let q(x) be the function defined in the statement of that lemma. The argument preceding Theorem 2.2 shows that the extension of q(x) to C + is the Stieltjes Transform of a positive measure. It results that q(x) is negative and increasing on (a, b). Let Ω = diag(ω 2 1 , . . . , ω 2 r ) where ω 2 k = -1/q(ρ k ). Then it is clear that function D(x) = det(q(x)Ω + I r ) has r roots in (a, b) which coincide with the ρ k . Theorem 2.3 will be proven if we find a sequence of matrices P n for which

H * (z) = q(z)Ω, i.e., Λ * (dt) = ½ E (t) Ω ν(dt).
Rearrange the elements of D n in such a way that all the d n j which belong to E are in the top left corner of this matrix. Let M n = [M n ij ] be a random ⌊nν(E)⌋ × r matrix with iid elements such that √ nM n 11 has mean zero and variance one. Let Z n be the n× r matrix obtained by adding n-⌊nν(E)⌋ rows of zeros below M n . Then the law of large numbers shows in conjunction with a normal family theorem argument that there is a set of probability one over which zm n (z)Z * n T n (z)Z n converges to q(z)I r uniformly on the compact subsets of (a, b). Consequently, there exists a sequence of deterministic matrices B n such that zm n (z)B * n T n (z)B n → q(z)I r uniformly on these compact subsets. Matrix

P n = A n B * n with A n = Ω 1/2 0 (N -r)×r
satisfies the required property. Theorem 2.3 is proven.

Proof of Corollary 2.1. Observe from Proposition 2.1 that (1+cm(z)t) -1 ν(dt) = -z lim(n -1 Tr T n (z)) = -czm(z) + 1 -c. Consequently, in this particular case, H * (x) is unitarily equivalent to -m(x) (cxm(x) -1 + c) Ω on (a, b).

Proof of the second order result

We start by briefly showing Proposition 2.2.

Proof of Proposition 2.2.

For any i = 1, . . . , p, it is clear that m(ρ i ) 2 > 0 and m ′ (ρ i ) > 0. An immediate calculus then gives m ′ (ρ i )∆(ρ i ) = m 2 (ρ i ) which shows that ∆(ρ i ) > 0.
To prove the second fact, we shall establish more generally that lim sup √ n H n (ρ i )+ g(ρ i )Ω < ∞. Invoking Equation (3) and its analogue m n (z) = (-z + t(1 + c n m n (z)t) -1 ν n (dt)) -1 , taking the difference and doing some straightforward derivations, we get that (m n (ρ i )-m(ρ i ))(∆(ρ i )+ε 1 ) = ε 2 where ε 1 → 0 and where |ε 2 | ≤ K/ √ n thanks to the first two items of Assumption 7. Hence

|m n (ρ i ) -m(ρ i )| ≤ K/ √ n. Now we have H n (ρ i ) + g(ρ i )Ω = m n (ρ i ) 1 + c n m n (ρ i )t - m(ρ i ) 1 + cm(ρ i )t Λ n (dt) + m(ρ i ) 1 + cm(ρ i )t Λ n (dt) - m(ρ i ) 1 + cm(ρ i )t ν(dt) × Ω.
which shows thanks to Assumption 7 that lim sup

√ n H n (ρ i ) + g(ρ i )Ω < ∞.
Proof of Theorem 2.4. In all the remainder of this section, we shall work on a sequence of factorizations P n = U n R * n such that Λ n satisfies the third item of Assumption 7. We also write

U n = [U 1,n • • • U t,n ] and R n = [R 1,n • • • R t,n ] where U i,n ∈ C N ×ji and R i,n ∈ C n×ji .
We now enter the core of the proof Theorem 2.4. The following preliminary lemmas are proven in the appendix: Lemma 5.1. Let s be a fixed integer, and let Z N = [Z ij ] be a N × s complex matrix with iid elements with independent N (0, 1/2) real and imaginary parts. Let Υ N = [Υ ij ] be a deterministic Hermitian N × N matrix such that Tr Υ N = 0, and let

F N = [F ij ] be a complex deterministic N ×s matrix. Assume that F * N F N → ς 2 I s , that lim sup N Υ N < ∞, and that N -1 Tr Υ 2 N → σ 2 as N → ∞.
Let M be a s × s complex matrix with iid elements with independent N (0, 1/2) real and imaginary parts, and let G be a s × s GUE matrix independent of M . Then

N -1/2 Z * N Υ N Z N , Z * N F N D ----→ N →∞ (σG, ςM ) . Lemma 5.2. For x ∈ supp(µ) c , E φ n ũ * n Q n (x)(n -1 Y * n Y n ) Q n (x)ũ n = c n x 2 m n (x) 2 ũ * n D n T 2 n (x)ũ n 1 -c n x 2 m n (x) 2 1 n Tr D 2 n T 2 n (x) + O(n -1 )
and

Var φ n ũ * n Q n (x)(n -1 Y * n Y n ) Q n (x)ũ n ≤ K n .
Lemma 5.3. For i = 1, . . . , p, let A i be a deterministic Hermitian j i × j i matrix independent of n, where p and the j i are as in the statement of Theorem 2.4. For i = 1, . . . , p, let M i,n a n × j i matrix such that sup n M i,n < ∞. Then for any

t ∈ R, E exp ı √ nt p i=1 Tr A i M * i,n ( Q n (ρ i )-T n (ρ i ))M i,n = exp(-t 2 σ2 n /2)+O(n -1/2 )
where

σ2 n = p i,k=1 c n ρ i ρ k m n (ρ i )m n (ρ k ) × Tr A i M * i,n T n (ρ i )D n T n (ρ k )M k,n A k M * k,n T n (ρ k )D n T n (ρ i )M i,n 1 -c n ρ i ρ k m n (ρ i )m n (ρ k ) 1 n Tr D n T n (ρ i )D n T n (ρ k )
.

Replacing the M i,n with the blocks R i,n of R n in the statement of Lemma 5.3 and observing that

R * n T n (ρ i )D n T n (ρ k )R n = t ρ i ρ k (1 + c n m n (ρ i )t)(1 + c n m n (ρ k )t) Λ n (dt),
we obtain from the third item of Assumption 7 that σ2 n → p i=1

σ2 i Tr A 2 i where σ2 i = cω 4 i ρ 2 i ∆(ρ i ) m(ρ i )t (1 + cm(ρ i )t) 2 ν(dt) 2 .
Invoking the Cramer-Wold device, this means that the p-uple of random matrices

√ n R * i,n ( Q n (ρ i ) -T n (ρ i ))R i,n p i=1 converges in distribution towards (σ i G i ) p i=1
where G 1 , . . . , G p are independent GUE matrices with G i ∈ C ji×ji . Lemmas 5.1-5.3 lead to the following result which plays a central role in the proof of Theorem 2.4:

Consider the 3p-uple of random matrices

L n = √ n× U * i,n Y n Q n (ρ i )R i,n √ n , U * i,n (Q n (ρ i ) -m n (ρ i )I N )U i,n , R * i,n ( Q n (ρ i ) -T n (ρ i ))R i,n p i=1
.

Define the following quantities

ς 2 i = ω 2 i ∆(ρ i ) m 2 (ρ i )t (1 + cm(ρ i )t) 2 ν(dt) σ 2 i = 1 ∆(ρ i ) m 4 (ρ i )t 2 (1 + cm(ρ i )t) 2 ν(dt) σ2 i = cω 4 i ρ 2 i ∆(ρ i ) m(ρ i )t (1 + cm(ρ i )t) 2 ν(dt) 2 .
Let M 1 , . . . , M p be random matrices such that M i ∈ C ji×ji and has independent elements with independent N (0, 1/2) real and imaginary parts.

Let G 1 , G 1 , . . . , G p , G p be GUE matrices such that G i , G i ∈ C ji×ji . Assume in addition that M 1 , G 1 , G 1 , . . ., M p , G p , G p are independent. Then L n D -→ n→∞ ς i M i , σ i G i , σi G i p i=1 . Proof. Let α n (ρ) = N -1 Tr Q n (ρ)
. By Lemmas 4.3 and 4.5, √ n(α n (ρ i )-m n (ρ i )) P → 0. Therefore, we can replace the m n (ρ i ) in the expression of L n by α n (ρ i ), as we shall do in the rest of the proof. Write s = j 1 + • • • + j p and let Z n be a N × s complex matrix with iid elements with independent N (0, 1/2) real and imaginary parts. Assume that Z n and X n are independent. Write

Z n = Z 1,n • • • Z p,n where the block Z i,n is N × j i . Let n -1/2 X n = W n ∆ n W *
n be a singular value decomposition of n -1/2 X n . By assumption 2, the square matrices W n and W n are Haar distributed over their respective unitary groups, and moreover, W n , ∆ n and W n are independent. Let

L n = √ n U * n Y n Q n (ρ i )R i,n √ n , U * n (Q n (ρ i ) -α n (ρ i )I N )U n , R * i,n ( Q n (ρ i ) -T n (ρ i ))R i,n p i=1 .
We have

L n L = √ N (Z * n Z n ) -1/2 Z * n F i,n , N 1/2 (Z * n Z n ) -1/2 Z * n Υ i,n Z n (Z * n Z n ) -1/2 , √ nR * i,n ( Q n (ρ i ) -T n (ρ i ))R i,n p i=1
where

F i,n = c -1/2 n ∆ n W * n D 1/2 n Q n (ρ i )R i,n and Υ i,n = c -1/2 n ((∆ n W * n D n W n ∆ n - ρ i ) -1 -α n (ρ i )I N ). We shall now show that the term √ N (Z * n Z n ) -1/2 Z * n F i,n can be replaced with Z * n F i,n
. By the law of large numbers, we have

N -1 Z * n Z n as → I s . By the independence of Z n and (∆ n , W n ), we have E[Tr Z * n F i,n F * i,n Z n | (∆ n W n )] = sc -1 n Tr R * i,n Q n (ρ i )(n -1 Y * n Y n ) Q n (ρ i )R i,n
whose limit superior is bounded with probability one. Hence Z * n F i,n is tight, proving that the replacement can be done. By deriving the variances of the elements of N -1/2 Z * n Υ i,n Z n with respect to the law of Z n , and by recalling that lim sup n Υ i,n is bounded with probability one, we obtain that these elements are also tight. It results that we can replace L n with

L n = Z * i,n F i,n , Z * i,n Υ i,n Z i,n √ N , √ nR * i,n ( Q n (ρ i ) -T n (ρ i ))R i,n p i=1
.

For i = 1, . . . , p, let A i and B i be deterministic Hermitian j i × j i matrices and let C i be deterministic complex j i × j i matrices, all independent of n. The lemma will be established if we prove that

E exp ı √ nt p i=1 Tr A i R * i,n ( Q n (ρ i ) -T n (ρ i ))R i,n ×E exp ıt p i=1 N -1/2 Tr B i Z * i,n Υ i,n Z i,n + ℜ(Tr C i Z * i,n F i,n ) (∆ n , W n ) ----→ n→∞ p i=1 exp(-t 2 (σ 2 i Tr A 2 i + σ 2 i Tr B 2 i + 1 2 ς 2 i Tr C i C * i )/2). ( 11
)
In addition to the boundedness of Υ i,n w.p. one, we have Tr Υ i,n = 0, and

1 N Tr Υ 2 i,n = 1 c n N N ℓ=1 (λ n ℓ -ρ i ) -1 -α n (ρ i ) 2 a.s. ----→ n→∞ c -1 (m ′ (ρ i ) -m(ρ i ) 2 ) = c -1 m(ρ i ) 2 ∆(ρ i ) -1 -1 = σ 2 i .
Moreover, using Lemma 5.2 in conjunction with Assumption 6, we obtain

F * i,n F i,n = c -1 n R * i,n Q n (ρ i ) 1 n Y * n Y n Q n (ρ i ) R i,n P ----→ n→∞ ς 2 i I ji .
From any sequence of integers increasing to infinity, there exists a subsequence along which this convergence holds in the almost sure sense. Applying Lemma 5.1, we get that the inner expectation at the left hand side of [START_REF] Capitaine | The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations[END_REF] converges almost surely along this subsequence towards

p i=1 exp(-t 2 (σ 2 i Tr B 2 i + 1 2 ς 2 i Tr C i C * i )/2
). Using in addition Lemma 5.3 along with the dominated convergence theorem, we obtain that Convergence [START_REF] Capitaine | The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations[END_REF] holds true along this subsequence. Since the original sequence is arbitrary, we obtain the required result.

The remainder of the proof of Theorem 2.4 is an adaptation of the approach of [START_REF] Benaych-Georges | Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices[END_REF].

Lemma 5.5. For a given x ∈ R and a given i ∈ {1, . . . , p}, let y = ρ i + n -1/2 x, and let

S n (y) = √ yU * n Q n (y)U n I r + n -1/2 U * n Y n Q n (y)R n I r + n -1/2 R * n Q n (y)Y * n U n √ yR * n Q n (y)R n χ (i) n (x) = n j i 2 det Ŝn (y) - t k =i ω 2 k g(ρ i ) -1 j k × det √ nU * i,n (Q n (ρ i ) -m n (ρ i )I N ) U i,n m(ρ i ) + ρ i m(ρ i ) √ nR * i,n ( Q n (ρ i ) -T n (ρ i ))R i,n -2ℜ U * i,n Y n Q n (ρ i )R i,n - √ n(H i,n (ρ i ) + I ji ) -xH ′ i,n (ρ i ) ( 12
)
where ℜ(M ) = (M + M * )/2 for a square matrix M . Then

(χ (i) n (x 1 ), . . . , χ (i) n (x p )) P ----→ n→∞ 0
for every finite sequence {x 1 , . . . , x p }.

Proof. We show the result for i = 1, the same procedure being valid for the other values of i. The notation X n = o P (1) means that the random variable X n converges to zero in probability, while ----

X n = O P (n -ℓ ) means that n ℓ X n is tight. Write U = [U 1 , Ū1 ] and R = [R 1 , R1 ] where Ū1 = [U 2 , . . . , U t ] and R1 = [R 2 , . . . , R t ]. Writing A = √ yU * 1 Q(y)U 1 √ yU * 1 Q(y) Ū1 √ y Ū * 1 Q(y)U 1 √ y Ū * 1 Q(y) Ū1 := A 11 A 12 A * 12 A 22 , B = I j1 + n -1/2 U * 1 Y Q(y)R 1 n -1/2 U * 1 Y Q(y) R1 n -1/2 Ū * 1 Y Q(y)R 1 I r-j1 + n -1/2 Ū * 1 Y Q(y) R1 := B 11 B 12 B 21 B 22 , and
C = √ yR * 1 Q(y)R 1 √ yR * 1 Q(y) R1 √ y R * 1 Q(y)R 1 √ y R * 1 Q(y
→ n→∞ √ ρ 1 m(ρ 1 )I r-j1 I r-j1 I r-j1 √ ρ 1 (cm(ρ 1 ) -ρ -1 1 (1 -c))Ω 2
which is invertible since det M 22 as → k>1 (ω 2 k g(ρ 1 ) -1) j k = 0. Moreover, M 12 = O P (n -1/2 ). To see this, consider for instance the term

√ nC 12 = √ nyR * 1 ( Q - T ) R1 + √ nyR * 1 T R1 .
= y √ nU * 1 (Q(y) -m n (y)I N )U 1 + y √ nm n (y)I j1 + √ n I j1 + U * 1 Y Q(y) √ n R 1 (R * 1 Q(y)R 1 ) -1 (R * 1 ( Q(y) -T (y))R 1 )(R * 1 T (y)R 1 ) -1 × I j1 + R * 1 Q(y)Y * √ n U 1 - √ n I j1 + U * 1 Y Q(y) √ n R 1 (R * 1 T (y)R 1 ) -1 I j1 + R * 1 Q(y)Y * √ n U 1 . (13)
To go further, remark that

y √ nm n (y)I j1 - √ n(R * 1 T (y)R 1 ) -1 = (R * 1 T (y)R 1 ) -1 √ n ym n (y)R * 1 T (y)R 1 -ρ 1 m n (ρ 1 )R * 1 T (ρ 1 )R 1 + √ n ρ 1 m n (ρ 1 )R * 1 T (ρ 1 )R 1 -I j1 = ρ 1 m(ρ 1 ) -xH ′ 1 (ρ 1 ) - √ n(H 1 (ρ 1 ) + I j1 ) + o(1)
where we that y

= ρ 1 + xn -1 2 → ρ 1 and that R * 1 T (y)R 1 → (ρ 1 m(ρ 1 )) -1 I j1 . Recall from Lemma 5.4 that √ nU * 1 (Q -m n I)U 1 , √ nR * 1 ( Q -T )R
= ρ 1 √ nU * 1 (Q(ρ 1 ) -m n (ρ 1 )I N )U 1 + (ρ 1 m(ρ 1 )) 2 √ nR * 1 ( Q(ρ 1 ) -T (ρ 1 )I n )R 1 -2ρ 1 m(ρ 1 )ℜ U * 1 Y Q(ρ 1 )R 1 -ρ 1 m(ρ 1 ) xH ′ 1 (ρ 1 ) + √ n(H 1 (ρ 1 ) + I j1 ) + o P (1).
Plugging this expression at the right hand side of the expression of n j1/2 det S and observing that H ′ 1,n (ρ 1 ) → -ω 2 1 g(ρ 1 ) ′ I j1 concludes the proof.

For i = 1, . . . , p, take x 1 (i) > y 1 (i) > x 2 (i) > y 2 (i) > . . . > y ji (i) fixed se- quences of real numbers. Call J n = ( √ n( λn k(i)+ℓ -ρ i ), i = 1, . . . , p, ℓ = 1, . . . , j i ), with k(i) = i-1 m=1 j m . Let also C be the rectangle C = [x 1 (1), y 1 (1)] × . . . × [x p (j p ), y p (j p )].
Then, for all large n, we have

P (J n ∈ C) = P det Ŝn ρ i + x ℓ (i) √ n det Ŝn ρ i + y ℓ (i) √ n < 0
since det Ŝn (t) changes sign around t = λn k(i)+ℓ , and only there (with probability one, for all large n). Lemma 5.5, we see that, for growing n, the probability for the product of the determinants above to be negative for all i and ℓ approaches the probability P det A x ℓ (i) det A y ℓ (i) < 0, i = 1, . . . , p, ℓ = 1, . . . , j i where A x is the matrix

A x = √ nU * i,n (Q n (ρ i ) -m n (ρ i )I N ) U i,n m(ρ i ) + ρ i √ nR * i,n ( Q n (ρ i ) -T n (ρ i ))R i,n ω 2 i (c + cρ i m(ρ i ) -1) -2ℜ U * i,n Y Qn (ρ i )R i,n - √ n(H i,n (ρ i ) + I ji ) -xH ′ i,n (ρ i ).
This last probability is equal to P( Jn ∈ C), where Jn is the vector obtained by stacking the p vectors of decreasingly ordered eigenvalues of the matrices

B i = [H ′ i,n (ρ i )] -1 √ nU * i,n (Q n (ρ i ) -m n (ρ i )I N ) U i,n m(ρ i ) + ρ i √ nR * i,n ( Q n (ρ i ) -T n (ρ i ))R i,n ω 2 i (c + cρ i m(ρ i ) -1) -2ℜ U * i,n Y Qn (ρ i )R i,n - √ n(H i,n (ρ i ) + I ji ) .
From Lemma 5.4, {B 1 , . . . , B t } asymptotically behave as scaled non-zero mean GUE matrices. Precisely, denoting Bi = H

′ i,n (ρ i )B i + √ n(H i,n (ρ i ) + I ji ), from Lemma 5.4 and for all a, b, E |( Bi ) ab | 2 → σ 2 i m(ρ i ) 2 + ρ 2 i σ2 i ω 4 i (c + cρ i m(ρ i ) -1) 2 + 2ς 2 = σ 2 i m(ρ i ) 2 + ρ 2 i m(ρ i ) 2 σ2 i + 2ς 2 = m 2 (ρ i ) ∆(ρ i ) t 2 ν(dt) (1 + cm(ρ i )t) 2 + cω 4 i m(ρ i )tν(dt) (1 + cm(ρ i )t) 2 2 + 2ω 2 i tν(dt) (1 + cm(ρ i )t) 2 .
This concludes the proof of Theorem 2.4.

N = N -1/2 Tr AZ * N Υ N Z N + ℜ(Tr BZ * N F N )
where ℜ(M ) = (M + M * )/2 for any square matrix M . We shall show that for any t ∈ R,

ϕ N (t) := E[exp(ıtΓ N )] ----→ N →∞ exp -t 2 σ 2 Tr A 2 + ς 2 Tr BB * /2 2 := exp - t 2 v 2 2 .
The result will follow by invoking the Cramér-Wold device. To establish this convergence, we show that the derivative ϕ ′ N (t) satisfies ϕ ′ N (t) = -tv 2 ϕ N (t) + ε N (t) where ε N (t) → 0 as N → ∞ uniformly on any compact interval of R. That being true, the function ψ

N (t) = ϕ N (t) exp(t 2 v 2 /2) satisfies ψ N (t) = 1 + t 0 ε N (u) exp(u 2 v 2 /2)du → 1 which proves the lemma.
By the IP formula, we get

ϕ ′ (t) = ıE[Γ exp(ıtΓ)] = ıE s i,j=1 N k,ℓ=1 A ij Z * kj Υ kℓ Z ℓi √ N + s i,j=1 N k=1 B ij Z * kj F ki + F * ki Z kj B * ij 2 × exp(ıtΓ) = ıE i,j,k,ℓ A ij Υ kℓ √ N ∂(Z ℓi exp(ıtΓ)) ∂Z kj + 1 2 i,j,k B ij F ki ∂ exp(ıtΓ) ∂Z kj + F * ki B * ij ∂ exp(ıtΓ) ∂Z * kj .
We obtain after a small calculation

∂ exp(ıtΓ) ∂Z kj = ıt [AZ * Υ] jk √ N + 1 2 [B * F * ] jk exp(ıtΓ), ∂ exp(ıtΓ) ∂Z * kj = ıt [ΥZA] kj √ N + 1 2 [F B] kj exp(ıtΓ) which leads to ϕ ′ (t) = -tE[N -1 Tr A 2 Z * Υ 2 Z exp(ıtΓ)] -(t/2) Tr(BB * F * F ) ϕ(t) + ıN -1/2 Tr A Tr Υ ϕ(t) -tE[N -1/2 Tr AB * F * ΥZ exp(ıtΓ)] -(t/2)E[N -1/2 Tr Z * ΥF BA exp(ıtΓ)].
Let us consider the first term at the right hand side of this equation. We have

E[N -1 Tr A 2 Z * Υ 2 Z] = N -1 Tr A 2 Tr Υ 2 .
Applying the Poincaré-Nash inequality, we obtain after some calculations that Var(N

-1 Tr A 2 Z * Υ 2 Z) ≤ 2N -2 Tr A 4 Tr Υ 4 = O(N -1 ) since Υ is bounded. It results that E[N -1 Tr A 2 Z * Υ 2 Z exp(ıtΓ)] = N -1 Tr A 2 Tr Υ 2 ϕ(t) + O(N -1/2
) by Cauchy-Schwarz inequality. The third term is zero by hypothesis. Finally,

N -1 E| Tr Z * ΥF BA| 2 = N -1 Tr Υ 2 F BA 2 B * F * ≤ N -1 Υ 2 Tr F BA 2 B * F * = O(N -1
). Hence, the last two terms are O(N -1/2 ) by Cauchy-Schwarz inequality, which proves Lemma 5.1.

A.2. An intermediate result.

The following lemma will be needed in the proof of Lemma 5.2:

Lemma A.1. For x, y ∈ supp(µ) c , E φ n 1 n Tr Q n (x)D Q n (y)D = 1 n Tr D T n (x)D T n (y) 1 -c n xm n (x)ym n (y) 1 n Tr D T n (x)D T n (y) + O(n -1 ) E φ n ũ * n Q n (x)D Q n (y)ṽ n = ũ * n T n (x)D T n (y)ṽ n 1 -c n xm n (x)ym n (y) 1 n Tr D T n (x)D T n (y) + O(n -1 ).
We denote here Q x = Q(x) and drop all unnecessary indices. Using the IP formula, we obtain

E φY * ia Y ij Q x,jp d p Q y,pq = d p d j n δ(a -j)E φ Q x,jp Q y,pq - 1 n E φ Q x,jj [Y Q x ] ip Y * ia Q y,pq - 1 n E φY * ia Q x,jp Q y,pj [Y Q y ] iq +E 1 n [adj(ψ)ψ ′ Y ] ij Y * ia Q x,jp Q y,pq .
Making the sum over i, p, and j, this is

1 n E φ[Y * Y Q x D Q y ] aq = 1 n 2 E [Y * adj(ψ)ψ ′ Y D Q x D Q y ] aq + c n d a E φ[ Q x D Q y ] aq - 1 n E φ 1 n Tr D Q x [Y * Y Q x D Q y ] aq - 1 n E φ 1 n Tr Q x D Q y D[Y * Y Q y ] aq .
Using the relation 1 n Y * Y Q x = x Q x + I n and appropriately gathering the terms on each side gives

E φ[ Q x D Q y ] aq (x -c n d a + x 1 n Tr D Q x ) = -E φ[D Q y ] aq (1 + 1 n Tr D Q x ) -E φ 1 n Tr Q x D Q y D(δ(a -q) + y[ Q y ] aq ) + E 1 n 2 [Y * adj(ψ)ψ ′ Y D Q x D Q y ] aq . (15)
Introducing the term βx = φ 1 n Tr D Q x and βx = βx -φE[ βx ], we have

E φ[ Q x D Q y ] aq (x -c n d a + xE[ βx ]) = -E φ[D Q y ] aq (1 + E[β x ]) -E φ 1 n Tr Q x D Q y D (δ(a -q) + yE[[ Q y ] aq ]) -E [D Q y ] aq βx -E [ Q x D Q y ] aq x βx -E φ 1 n Tr Q x D Q y Dy([ Q y ] aq -E[ Q y ] aq ]) + E 1 n 2 [Y * adj(ψ)ψ ′ Y D Q x D Q y ] aq . (16)
At this point, we can prove both results for the trace and for the quadratic form. We start by dividing each side by xc n d a + xE[ βx ]. We begin with the trace result. Multiplying the resulting left-and right-hand sides by d a , summing over a = q and normalizing by 1/n, we obtain

E φ 1 n Tr Q x D Q x D = -(1 + E[ βx ])E φ 1 n Tr D Q y DA x -E φ 1 n Tr Q x D Q y D yE[φ 1 n Tr DA x Q y ] + 1 n Tr DA x + ε n
where we denoted A x = (x(1 + E[ βx ])I nc n D) -1 and where

ε n = E Tr Y * adj(ψ)ψ ′ Y n 3 D Q x D Q y DA x -E 1 n Tr D Q y DA x βx -E 1 n Tr Q x D Q y DA x βx -E φ 1 n Tr Q x D Q y Dy 1 n Tr Q y DA x -E( 1 n Tr Q y DA x ) . (17)
From Lemma 4.5, E[ βx ] = δx + O(n -2 ), where we denoted δx = 1 n Tr D T x . Also, it is easily observed that

I n (1 + δx )x -c n D -1 = - 1 1 + δx T x (18)
with T x = T (x). Therefore, along with Lemma 4.5, we now have It therefore remains to prove that ε n = O(n -1 ). Due to the presence of φ in the expression of βx , and using Lemma 4.3 and Cauchy-Schwarz inequality, one can see that the last three terms in the expression of ε n are O(n -1 ). As for the first term, it is treated in a similar manner as in the proof of Lemma 4.6, and is O(n -2 ).

E φ 1 n Tr Q x D Q x D = 1 n Tr D T x D T y + E φ 1 n Tr Q x D Q x D
In order to prove the result on the quadratic form, we start again from [START_REF] Girko | of Mathematics and its Applications[END_REF]. Dividing each side again by xc n d a + xE[ βx ], introducing [ũ] a , [ṽ] q , and summing over the indices, we obtain

E φũ * Q x D Q y ṽ = -E φũ * A x D Q y ṽ -E φ 1 n Tr Q x D Q y D ũ * A x (yE[φ Q y ] + I n )ṽ + ε ′ n (19)
where ε ′ n is very similar to ε n and is shown to be O(n -1 ) with the same line of arguments. Using Lemma 4. We provide here its main steps. By the IP formula, we have

E[φ Q pk Y * ℓk Y ℓm Q mr ] = - d m n E[φ Q pk Y * ℓk Q mm [Y Q] ℓr ] + δ(k -m)d m E[φ Q pk Q mr ] - d m n E[φY * ℓk Q mr Q pm [Y Q] ℓk ] + d m n E[ Q pk Y * ℓk Q mr [adj(ψ)ψ ′ Y ] ℓm ]
Taking the sum over m, we obtain

E[φ Q pk Y * ℓk [Y Q] ℓr ] = d k 1 + E[ β] E[φ Q pk Q kr ] - 1 1 + E β 1 n E[φY * ℓk [ QD Q] pr [Y Q] ℓk ] + 1 1 + E[ β] 1 n E[ Q pk Y * ℓk [adj(ψ)ψ ′ Y D Q] ℓr ] -E[ β Q pk Y * ℓk [Y Q] ℓr ]
where β(x) = φ 1 n Tr D Q(x) and β(x) = β(x)-φE[ β(x)] as in the proof of Lemma A.1. Taking the sum over ℓ then over k, we obtain

E[φ[ Q Y * Y n Q] pr ] = c n 1 1 + E[ β] E[φ[ QD Q] pr ] - 1 1 + E[ β] E[φ[ QD Q] pr 1 n Tr( Y * Y n Q)] + 1 1 + E[ β] 1 n E[[ Q Y * adj(ψ)ψ ′ Y n D Q] pr ] -E[ β[ Q Y * Y n Q] pr ]
Observing

Y * Y n Q j M j M * j Q j D Q 1   + ε ′ n with A ρi = (ρ i (1 + E[ βi ])I n -c n D) -1
, and

ε ′ n = Tr M * 1 A ρ1 E ′ M 1
with (E ′ ) pq = ε ′ pq . From [START_REF] Hachem | A new approach for mutual information analysis of large dimensional multi-antenna chennels[END_REF], the identity n -1 Y * Y Q j = I n +ρ j Q j , and Lemma 4.

1 . Introduction 1 . 1 .

 111 The model and the literature. Consider a sequence of N × n matrices Y n , n = 1, 2, . . ., of the form Y n = X n D 1/2 n

( 2 )

 2 The distances d(d n j , supp(ν)) from d n j to supp(ν) satisfy max j∈{1,...,n} d d n j , supp(ν) ----→ n→∞ 0.

Assumption 5 .

 5 Any two accumulation points Λ 1 and Λ 2 of the sequences Λ GS n satisfy Λ 1 (dx) = W Λ 2 (dx)W * where W is a r × r unitary matrix.

Theorem 2 . 2 .

 22 Let Assumptions 1, 2 and 3 hold true. Denote by ( λn i ) i=1,...,N the eigenvalues of Σ n Σ * n . Let (a, b) be any connected component of supp(µ) c = Rsupp(µ). Then the following facts hold true:

( 1 )

 1 Let (P n ) be a sequence satisfying Assumptions 4 and 5. Given an accumulation point Λ * of a sequence Λ n , let H * (z) = m(z)(1 + cm(z)t) -1 Λ * (dt). Then H * (z) can be analytically extended to (a, b) where its values are Hermitian matrices, and the extension is increasing in the order of Hermitian matrices on (a, b). The function D(x) = det(H * (x) + I r ) has at most r zeros on (a, b). Let ρ 1 , . . . , ρ k , k ≤ r be these zeros counting multiplicities. Let I be any compact interval in (a, b) such that {ρ 1 , . . . , ρ k } ∩ ∂I = ∅. Then ♯{i : λn i ∈ I} = ♯{i : ρ i ∈ I} with probability 1 for all large n. (2) Let A = inf (supp(µ) -{0}). Then for any positive A ′ < A (assuming it exists) and for any sequence of matrices (P n ) satisfying Assumption 4,

Corollary 2 . 1 .

 21 Assume the setting of Theorem 2.2-(1), and let Assumption 6 hold true. Then the function g(x) = m(x) (cxm(x) -1 + c) is decreasing on (a, b).

3. 3 .

 3 Differentiation formulas. Let ∂/∂z = (∂/∂x-ı∂/∂y)/2 and ∂/∂ z = (∂/∂x+ ı∂/∂y)/2 for z = x + ıy. Given a Hermitian matrix X with a spectral decomposition

3. 5 .

 5 Analysis of the support of µ. Proposition 3.1 ([37], Th.1.1). For all x ∈ R * , lim z∈C+→x m(z) exists. The limit that we denote m(x) is continuous on R * . Moreover, µ has a continuous density f on R * given by f (x) = π -1 ℑm(x).

Proposition 3 . 2 (

 32 [START_REF] Silverstein | Analysis of the limiting spectral distribution of largedimensional random matrices[END_REF], Th. 4.1 and 4.2). For any x 0 ∈ supp(µ) c , let m 0 = m(x 0 ). Then m 0 ∈ B, x 0 = x(m 0 ), and x ′ (m 0 ) > 0. Conversely, let m 0 ∈ B such that x ′ (m 0 ) > 0. Then x 0 = x(m 0 ) ∈ supp(µ) c , and m(x 0 ) = m 0 .

  2 for z ∈ I, we deduce that |m(z)| and f (z) = d(-(cm(z)) -1 , supp(ν)) are positive on D I . Since these functions are continuous on this compact set, min |m(z)| = K 1 > 0 and min f (z) = K 2 > 0 on D I . Consequently, for any z ∈ D I and any t ∈ supp(ν), |1 + cm(z)t| = |cm(z)(-(cm(z)) -1t)| ≥ |cm(z)|f (z) ≥ cK 1 K 2 > 0. We now prove the second inequality. Denote by d H (A, B) the Hausdorff distance between two sets A and B.

Lemma 3 . 2 .

 32 ν)) similarly. Since m n (z) converges uniformly to m(z) and inf |m(z)| > 0 on D I , we get that d(-(c n m n (z)) -1 , -(cm(z)) -1 ) → 0 uniformly on this disk. By Assumption 3, d H (supp(ν n ), supp(ν)) → 0. Hence f n (z) converges uniformly to f (z) on D I which proves the second inequality. Assume the setting of Lemma 3.1, and assume that {0} ∈ I. Then for any sequence of vectors ũn ∈ C n such that sup n ũn < ∞, the quadratic forms ũ * n T n (z)ũ n are the Stieltjes Transforms of positive measures γ n such that sup n γ n (R) < ∞ and γ n (I) = 0 for n large enough.

  3 and Prop.5] from z ∈ (-∞, 0) to z ∈ C + . The following important result, due to Haagerup and Thorbjørnsen, is established in the proof of [17, Th.6.2]: Lemma 3.3. Assume that h(z) is an analytic function on C + that satisfies |h(z)| ≤ P (|z|)R(|ℑ(z)| -1 ) where P and R are polynomials with nonnegative coefficients. Then for any function ϕ ∈ C ∞ c (R, R), the set of smooth real-valued functions with compact support in R, lim sup y↓0 R ϕ(x)h(x + ıy)dx < ∞.

Lemma 4 . 5 ,

 45 we have α = c n m n + O(n -2 ). Using in addition the bound[START_REF] Benaych-Georges | The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices[END_REF], weobtain α = n -1 Tr(D(-z(I + c n m n D + (αc n m n )D) -1 ) = n -1 Tr D T + O(n -2 ). Since m n (z) = (-z(1+n -1 Tr D T (z))) -1 , we obtain that (-z(1 + α)) -1 = m n (z)+ O(n -2). Using in addition Lemma 4.4, we obtain that the first right hand side term of (9) is u * v m n (z) + O(n -2 ). Due to the presence of φ in the expression of β, the second term is bounded by KE| β|. Moreover,β = n -1 φ Tr Qn -1 E[φ Tr Q] + (1φ)n -1 E[φ Tr Q]. ByLemmas 4.4 and 4.3, E| β| = O(n -1 ). The third term can be shown to be bounded by Kn -1 E Tr ϕ(n -1 Y Y * ) = O(n -2 ) where ϕ is as in the statement of Lemma 4.1. This proves the first inequality in the statement of the lemma.

  • I | det(H * (z)+I r )| > 0 which implies that lim inf n inf z∈∂D • I | det S n (z)| > 0. It follows that L n = L n and L n = L for n large enough. This concludes the proof of Theorem 2.2-(1). Proofs of Theorems 2.2-(2) and 2.3. We start with the following lemma: Lemma 4.8. Let A = inf(supp(µ)-{0}) and let (a, b) be a component of supp(µ) c . Then the following facts hold true:

  x 0 ] ⊂ supp(µ) c , the image of this interval by m belongs to B. By Proposition 3.4, lim x→0 + m(x) =

Figure 1 .

 1 Figure 1. Plot of x(m) for c = 0.1 and ν = 0.5(δ 1 + δ 3 ). The thick segment represents supp(µ).

Figure 2 .

 2 Figure 2. Plot of x(m) for c = 5 and ν = 0.5(δ 1/2 + δ 5/2 ). The thick segments represent supp(µ).

Appendix A . 3 A. 1 .

 .31 Proofs of Lemmas 5.1 to 5.Proof of Lemma 5.1. Given a s × s deterministic Hermitian matrix A and a s × s deterministic complex matrix B, let Γ

y 1 n 1 n 1 n

 111 Tr D T x T y + 1 n Tr D T x 1 + δx + ε n + O(n -2 ).Using now the fact that y T y +I n = c n 1 1+ δy D T y , we conclude E φ Tr Q x D Q x D = Tr D T x D T y 1c n (1 + δx ) -1 (1 + δy ) -1 1 n Tr D T x D T y + ε n + O(n -2 ).

A. 3 .

 3 3, (18), and the previous result onE[φ 1 n Tr Q x D Q y D],we finally obtainE φũ * Q x D Q y ṽ = ũ * T x D T y ṽ 1 + c n (1 + δx ) -1 (1 + δy ) -1 1 n Tr D T x D T y 1c n (1 + δx ) -1 (1 + δy ) -1 1 n Tr D T x D T y + O(n -1 ).from whichE φũ * Q x D Q y ṽ = ũ * T x D T y ṽ 1c n (1 + δx ) -1 (1 + δy ) -1 1 n Tr D T x D T y + O(n -1 ).conclude with the remark xm n (x) = -(1 + δx ) -1 . Proof of Lemma 5.2. The line of proof closely follows the proof of Lemma A.1.

A. 4 .=

 4 that (1 + E[ β(x)]) -1 = -xm n (x) + O(n -2 ) and making the usual approximations, we getE[φũ * Q Y * Y n Qũ] = xm n (x) 1 n Tr(E[φ Y * Y n Q])c n xm n (x) E[φũ * QD Qũ] + O(n -1 ) Observing that n -1 Tr(E[φ(n -1 Y * Y ) Q(x)] = N n -1 xm n (x) + N n -1 + O(n -2) and invoking Lemma A.1, we obtain the desired result. Proof of Lemma 5.3. As in the previous proofs, we discard unnecessary indices. We also denoteQ i = Q(ρ i ).For readability, we also write Mi = M i,n A i and use the shortcut notation Γ = √ n p i=1 Tr M * i Q i Mi . We focus first on the term in ρ 1 . The line of proof closely follows that of Lemma 5.1 with the exception that we need to introduce the regularization function φ to ensure the existence of all the quantities under study. That is, with ϕ N (t) = E[exp(ıtφΓ)], we only need to show that ϕ′ N (t) = -tσ 2 n ϕ N (t) + O(1/ √ n). Using |ϕ N (t)| ≤ 1 and Lemma 4.4, |E[exp(ıtΓ)]ϕ N (t)| ≤ 1 -E[φ] → 0 as N → ∞, from which the result unfolds.Using the IP formula, we first obtainc n E φ[D Q 1 ] pq e ıtφΓ -+ ε n,pq where ε n,pq = E 1 n Y * adj(ψ)ψ ′ Y n D Q 1 pq e ıtφΓ + E φ 1 n Y * adj(ψ)ψ ′ Y n D Q 1 pq ıtΓe ıtφΓand where we denoted X a the column a of matrix X, X * a being the row vector (X a ) * .With βj = φ 1 n Tr D Q j , βj = βj -φE βj , and with the relationn -1 Y * Y Q 1 = I n + ρ 1 Q 1 , we obtain ρ 1 (1 + E[ β1 ])c n d p E φ[ Q 1 ] pq e ıtφΓ = -δ(pq)(1 + E[ β1 ])E φe ıtφΓ-Dividing each side by ρ 1 (1 + E[ β1 ])c n d p , then multiplying by ( M1 ) p and (M 1 ) q , and summing over p, q gives E[φ Tr( M * 1 Q 1 M 1 )e ıtφΓ ] = -(1 + E[ β1 ])E[φe ıtΓ ] Tr M * 1 A ρ1 M 1

1 =

 1 5, we finally obtainE φ Tr( M * 1 Q 1 M 1 )e ıtφΓ -E[φe ıtφΓ ] Tr M1 T 1 M ıtE[φe ıtφΓ ] * j T j D T 1 M 1 1c n (1 + δ1 ) -1 (1 + δj ) -1 1 n Tr D T 1 D T j + ε ′ n + O(n -2 )

  Tr U Q(z) is the Stieltjes Transform of a positive measure τ n such that sup n τ n (R) < ∞ and supp(τ n ) ⊂ supp(θ n ) ∪ {0}. By Lemma 3.2, n -1 Tr U T (z) is the Stieltjes Transform of a positive measure γ n such that sup n γ n (R) < ∞ and γ n (I) = 0 for all large n. With the help of the second inequality of Proposition 3.5, we have a result similar to that of Proposition 3.6, namely that |E ϕdτ nϕdγ n | ≤ K/n 2 for any function ϕ ∈ C ∞ c (R, R). We can then prove the second inequality similarly to the first one.

	t) t -z	µ n (dt) ≤	K n 2
	by Proposition 3.3, and the first inequality is proved.
	By performing a spectral factorization of n -1 Y Lemma 4.6. The following inequalities hold true:

* Y , one can check that n -1

  2-(1). this interval in the order of Hermitian matrices, and the properties of this function given in the statement of Theorem 2.2-(1) are established.We now prove the convergence stated in Theorem 2.2-(

	Proof of Theorem 2.2-(1). Our first task is to establish the properties of H * (z)
	given in the statement of Theorem 2.2-(1). By Lemma 3.1 and the fact that
	supp(Λ

* ) ⊂ supp(ν), the function H * (z) can be analytically extended to (a, b). The comments preceding Theorem 2.2 show that any H * (z) is the Stieltjes Transform of a matrix-valued nonnegative measure Γ. Since H * (z) is analytic on (a, b), it is increasing on

  a row and column permutation. Hence n j1/2 det S = det M 22 ×n j1/2 det(M 11 -M 12 M -1 22 M * 12 ). Write Ω = diag(ω 2 1 I j1 , Ω 2 ).From the first order analysis we get that

				) R1	:=	C 11 C 12 C * 12 C 22	,
	we have det S = det	A B B * C	= det	      B * A 11 B 11 11 C 11 A * 12 B 21 B * 12 C * 12	A 12 B 12 B * 21 C 12 22 C 22 B * A 22 B 22	     	:= det	M 11 M 12 M * 12 M 22
	M 22	a.s.

after

  The first term is tight by Lemma 5.4 while the second is bounded by Assumption 7. The other terms are treated similarly. It results that M 12 M -1 22 M * 12 = O P (1/n). In addition, det(y -1/2 C 11 )

	definition of ρ 1 . From these observations we get that
	n j1/2 det S =	k>1 (ω 2 k g(ρ 1 ) -1) j k + o P (1) (ρ 1 m(ρ 1 )) -j1 + o P (1)
			× det	√ nyA 11 -	√ nyB 11 C -1 11 B * 11 + O P (n -1/2 ) .
	Now we make the expansion
	√ nyA 11 -	√ nyB 11 C -1 11 B * 11
			as → [ω 2 1 (cm(ρ 1 ) -ρ -1 1 (1 -c))] j1 = (ρ 1 m(ρ 1 )) -j1 by the

The work of the first three authors was supported by the French Ile-de-France region, DIM LSC fund, Digiteo project DESIR. 1

T i = T (ρ i ), from which

It remains to show that ε ′ n = O(n -1 ). We have explicitly

Using the fact that |e ıtφΓ | = 1 and the relation n