Damien Passemier
email: damien.passemier@gmail.com

Matthew R Mckay
email: eemckay@ust.hk

Yang Chen
email: yayangchen@umac.mo

Asymptotic Linear Spectral Statistics for Spiked Hermitian Random Matrix Models

Keywords: random matrix theory, high-dimensional statistics, spiked population model, Wishart distribution, F -matrix, MIMO systems, hypothesis testing

Using the Coulomb Fluid method, this paper derives central limit theorems (CLTs) for linear spectral statistics of three "spiked" Hermitian random matrix ensembles. These include Johnstone's spiked model (i.e., central Wishart with spiked correlation), non-central Wishart with rank-one non-centrality, and a related class of non-central F matrices. For a generic linear statistic, we derive simple and explicit CLT expressions as the matrix dimensions grow large. For all three ensembles under consideration, we find that the primary effect of the spike is to introduce an O(1) correction term to the asymptotic mean of the linear spectral statistic, which we characterize with simple formulas. The utility of our proposed framework is demonstrated through application to three different linear statistics problems: the classical likelihood ratio test for a population covariance, the capacity analysis of multi-antenna wireless communication systems with a line-of-sight transmission path, and a classical multiple sample significance testing problem.

Introduction

In multivariate analysis, many statistics of interest can be written as a sum of functions of eigenvalues of a sample covariance matrix or an F matrix. These are often referred to as "linear spectral statistics". In classical settings, the asymptotic distribution of such statistics has been studied extensively, typically assuming that the sample size m is large whilst the data dimension n is fixed. Modern applications, however, are often characterized by high-dimensional data sets, with m and n comparable. Representative examples include financial portfolios involving large numbers of assets [START_REF] Ledoit | Honey, I shrunk the sample covariance matrix[END_REF][START_REF] Rubio | Performance analysis and optimal selection of large minimum variance portfolios under estimation risk[END_REF], biomedical data sets involving large nucleotide or protein arrays [START_REF] Dahirel | Coordinate linkage of HIV evolution reveals regions of immunological vulnerability[END_REF][START_REF] Quadeer | Statistical linkage of mutations in the non-structural proteins of Hepatitis C virus exposes targets for immunogen design[END_REF], or modern wireless communication signals with large numbers of antennas [START_REF] Hoydis | Massive mimo in the ul/dl of cellular networks: How many antennas do we need?[END_REF][START_REF] Matthaiou | Sum rate analysis of zf receivers in distributed mimo systems[END_REF]. In such cases, statistical results based on classical asymptotic analysis are no longer accurate or meaningful. Motivated by the above considerations, there has been much recent interest in evaluating the asymptotic distribution of linear statistics for high-dimensional data models. Such models have been studied using ideas from random matrix theory to evaluate the asymptotic distribution under "double asymptotics", in which the data dimension and number of samples are both large and of similar order. In particular, this has led to the derivation of central limit theorems (CLTs) for linear spectral statistics of sample covariance matrices in [START_REF] Chen | On the linear statistics of Hermitian random matrices[END_REF][START_REF] Lytova | Central limit theorem for linear eigenvalue statistics of random matrices with independent entries[END_REF] , as well as for various other random matrix ensembles (see, e.g., [START_REF] Diaconis | Linear functional of eigenvalues of random matrices[END_REF][START_REF] Zheng | Central limit theorems for linear spectral statistics of large dimensional F -matrices[END_REF]). These results provide generic asymptotic formulas for the mean and variance of the limiting Gaussian distribution, and have been utilized for various applications (see e.g., [START_REF] Bai | Corrections to LRT on large-dimensional covariance matrix by RMT[END_REF][START_REF] Chen | Coulumb fluid, Painlevé transcendents, and the information theory of MIMO systems[END_REF][START_REF] Bai | Testing linear hypotheses in high-dimensional regressions[END_REF], among others).

Thus far, most results along this line assume that the population covariance matrix is the identity (e.g., [START_REF] Chen | On the linear statistics of Hermitian random matrices[END_REF][START_REF] Lytova | Central limit theorem for linear eigenvalue statistics of random matrices with independent entries[END_REF][START_REF] Anderson | A CLT for a band matrix model[END_REF]). In hypothesis testing problems, this allows one to capture information under the null hypothesis, but not the alternative. A more general model, introduced by Johnstone [START_REF] Johnstone | On the distribution of the largest eigenvalue in principal components analysis[END_REF], is the so-called "spiked" model, for which the population covariance matrix has all of its eigenvalues equal, except for a fixed few (referred to as the spike eigenvalues). Such models have attracted considerable attention. A major focus thus far has been on characterizing the statistical behavior of the extreme eigenvalues [START_REF] Baik | Phase transition of the largest eigenvalue for non null 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][START_REF] Bai | Central limit theorems for eigenvalues in a spiked population model[END_REF][START_REF] Rao Nadakuditi | Fundamental limit of sample generalized eigenvalue based detection of signals in noise using relatively few signal-bearing and noiseonly samples[END_REF][START_REF] Bai | On sample eigenvalues in a generalized spiked population model[END_REF], and this has found various applications in finance, signal processing, wireless communication and networking, to name a few (see [START_REF] Rao Nadakuditi | Sample eigenvalue based detection of highdimensional signals in white noise using relatively few samples[END_REF][START_REF] Torun | Portfolio risk in multiple frequencies[END_REF][START_REF] Couillet | Random Matrix Methods for Wireless Communications[END_REF][START_REF] Bianchi | Performance of statistical tests for singlesource detection using random matrix theory[END_REF][START_REF] Couillet | Fluctuations of spiked random matrix models and failure diagnosis in sensor networks[END_REF] and references therein). In contrast to the extreme eigenvalues, there has been considerably less work dealing with linear spectral statistics of spiked matrix models. A key exception is the very recent work [START_REF] Wang | A note on the CLT of the LSS for sample covariance matrix from a spiked population model[END_REF] which derived a CLT expression for linear spectral statistics for Johnstone's spiked model, based on employing the results from1 [START_REF] Bai | CLT for linear spectral statistics of large-dimensional sample covariance matrices[END_REF]. That result is given in terms of contour integrals. The results in [START_REF] Wang | A note on the CLT of the LSS for sample covariance matrix from a spiked population model[END_REF] have been subsequently applied to some specific linear spectral statistics in [START_REF] Passemier | Variance estimation and goodness-of-fit test in a highdimensional strict factor model[END_REF][START_REF] Wang | On the sphericity test with large-dimensional observations[END_REF].

In addition to Johnstone's spiked model, it turns out that alternative random matrix models exist which have close analogies. These include non-central Wishart matrices with rank-one non-centrality parameter (representing a spike), and a related class of non-central F matrices. In this paper, we will deal with all three classes of matrices, which we refer to collectively as spiked ensembles. We focus on basic models with a single spike, and with a single linear statistic. Natural extensions to account for multiple linear statistics (e.g., problems of the type considered in [START_REF] Zheng | Central limit theorems for linear spectral statistics of large dimensional F -matrices[END_REF][START_REF] Bai | CLT for linear spectral statistics of large-dimensional sample covariance matrices[END_REF]) and also to account for multiple spikes (such as the models considered in [START_REF] Wang | A note on the CLT of the LSS for sample covariance matrix from a spiked population model[END_REF]) are interesting and non-trivial, and these will be considered in future work. For each class, we derive new general CLT formulas for arbitrary linear statistics. In all three cases, we demonstrate that the effect of the spiked eigenvalue is to induce an O(1) correction term to the mean of the asymptotic Gaussian distribution, whilst not affecting the leading order terms of either the mean or variance. These results are consistent with previous phenomena observed in [START_REF] Wang | A note on the CLT of the LSS for sample covariance matrix from a spiked population model[END_REF] for Johnstone's spiked model. For each of the three models under consideration, we explicitly characterize the correction term via a remarkably simple formula involving only a single basic integral, which may be solved for any given linear statistic of interest.

To highlight the utility of our general results, we provide three representative example applications, one for each matrix model. For Johnstone's spiked correlation model, we examine a classical likelihood ratio test (LRT) statistic for the population covariance. Through our framework, we extract a known CLT result derived recently in [START_REF] Wang | A note on the CLT of the LSS for sample covariance matrix from a spiked population model[END_REF][START_REF] Onatski | Asymptotic power of sphericity tests for highdimensional data[END_REF]. This is achieved very efficiently (in the manner of a few lines), in contrast with the derivation in [START_REF] Onatski | Asymptotic power of sphericity tests for highdimensional data[END_REF], which relied on sophisticated tools of contiguity and Le Cam's lemmas (see [START_REF] Van Der | Asymptotic Statistics, Cambridge Series in Statistical and Probabilistic Mathematics[END_REF]). Furthermore, it serves as an alternative of the calculation in [START_REF] Wang | A note on the CLT of the LSS for sample covariance matrix from a spiked population model[END_REF]. For the non-central Wishart and non-central matrix F models, we present new results using our framework. In the first case, we consider the mutual information of multiple-input multiple-output (MIMO) wireless communication systems with a direct line-of-sight (LoS) transmission path, and derive a new CLT expression for the asymptotic distribution of this quantity with large numbers of antennas. For the non-central F model, a CLT is derived for a classical multiple sample significance test with high-dimensional data, under an appropriate alternative hypothesis. This new result is complementary to the recent result in [START_REF] Bai | Testing linear hypotheses in high-dimensional regressions[END_REF], which derived a corresponding CLT under the null.

The derivations in this paper are based on the Coulomb Fluid approach of random matrix theory. This approach was originally introduced by Dyson [START_REF] Dyson | Statistical theory of energy levels of complex systems i-iii[END_REF], and has been used extensively among the mathematical physics community for deriving large dimensional asymptotics of various random matrix ensembles (see e.g., [START_REF] Chen | On the linear statistics of Hermitian random matrices[END_REF][START_REF] Chen | Asymptotic level spacing of the Laguerre ensemble: A Coulomb fluid approach[END_REF][START_REF] Chen | Distribution of linear statistics in random matrix models (metallic conductance fluctuations)[END_REF][START_REF] Chen | Thermodynamic relations of the Hermitian matrix ensembles[END_REF][START_REF] Basor | Perturbed Hankel determinants[END_REF][START_REF] Simon | Crossover from conserving to lossy transport in circular random-matrix ensembles[END_REF][START_REF] Vivo | Large deviations of the maximum eigenvalue in Wishart random matrices[END_REF][START_REF] Vivo | Distributions of conductance and shot noise and associated phase transitions[END_REF][START_REF] Dean | Extreme value statistics of eigenvalues of Gaussian random matrices[END_REF][START_REF] Katzav | Large deviations of the smallest eigenvalue of the Wishart-Laguerre ensemble[END_REF]). Such tools have also recently found use in the information theoretic analysis of wireless communication systems [START_REF] Chen | Coulumb fluid, Painlevé transcendents, and the information theory of MIMO systems[END_REF][START_REF] Kazakopoulos | Living at the edge: A large deviations approach to the outage MIMO capacity[END_REF][START_REF] Li | On the distribution of MIMO mutual information: An in-depth Painlevé based characterization[END_REF][START_REF] Chen | Random matrix models, double-time Painlevé equations, and wireless relaying[END_REF]. Most relevant to the current paper is the work of Chen and Lawrence [START_REF] Chen | On the linear statistics of Hermitian random matrices[END_REF], which applied the Coulomb Fluid approach to derive CLTs for linear spectral statistics of classical random matrix ensembles, in the absence of spiked eigenvalues. To our knowledge, prior to the current work, such tools had yet to be applied to spiked random matrix models.

In this paper, we demonstrate that the Coulomb Fluid approach can be naturally applied for spiked random matrix ensembles, upon expressing the joint eigenvalue densities of the ensembles via convenient contour integral representations. For Johnstone's spiked model and the non-central Wishart model, such representations were discovered recently in [START_REF] Onatski | Asymptotic power of sphericity tests for highdimensional data[END_REF][START_REF] Wang | The largest eigenvalue of real symmetric, Hermitian and Hermitian self-dual random matrix models with rank one external source, part i[END_REF][START_REF] Mo | Rank 1 real Wishart spiked model[END_REF] and [START_REF] Dharmawansa | Three problems related to the eigenvalues of complex non-central Wishart matrices with rank-1 mean[END_REF] respectively, whilst for the matrix F model, we derive such a representation in the current paper, which also constitutes a new result.

Notation. All columns vectors and matrices are denoted by lowercase and uppercase boldface characters respectively. The conjugate transpose of a matrix A is A † . I n is the identity matrix of size n × n, whereas 0 n×m is the n × m matrix of all zeros. E(X) denotes the expectation of the random variable X. CW n (m, Σ, Θ) denotes the complex Wishart distribution of size n with m degrees of freedom, scale matrix Σ and non-centrality matrix Θ. N (µ, σ 2 ) denotes the Gaussian distribution with mean µ and variance σ 2 , whereas CN (u, Σ) denotes the circularly-symmetric complex Gaussian distribution with mean u and covariance matrix Σ. We use L → to denote convergence in distribution, and P to denote Cauchy principal value when dealing with principal value integrals.

Matrix Models and Eigenvalue Distributions

We consider the following three "spiked" random matrix models:

• Model A: Spiked central Wishart:
Matrices with distribution CW n (m, Σ, 0 n×n ) (m ≥ n), where Σ has one "spike" eigenvalue equal to 1 + δ with δ ≥ 0, and all other eigenvalues equal to 1.

• Model B: Spiked non-central Wishart:

Matrices with distribution CW n (m, I n , Θ) (m ≥ n),
where Θ is rank 1 (or zero) with "spike" eigenvalue nν for ν ≥ 0.

• Model C: Spiked multivariate F: Matrices of the form

F = W 1 W -1 2 , where W 1 ∼ CW n (m 1 , Σ, Θ) (m 1 > n), W 2 ∼ CW n (m 2 , Σ, 0 n×n ) (m 2 > n) are independent, with Θ rank 1 (or zero) having "spike" eigenvalue nν for ν ≥ 0.
For these three models, expressions for the joint probability density functions of the eigenvalues x k , 1 ≤ k ≤ n (taken in the following to be unordered) are well-known in various forms; for example, in terms of zonal polynomials [START_REF] James | Distributions of matrix variates and latent roots derived from normal samples[END_REF] or a determinant [START_REF] Mehta | Random Matrices, 3rd Edition, Pure and Applied Mathematics[END_REF]. Quite recently, however, it has been discovered that for Models A and B, the eigenvalue densities admit a particularly convenient contour integral representation

K n [l] 2πı C l(z) 1≤j<k≤n (x k -x j ) 2 n j=1 x m-n j e -x j z -x j dz, (1)
for x j ∈ (0, ∞), 1 ≤ j ≤ n, where K n [l] is a normalization constant, and the contour C encloses counter-clockwise x 1 , . . . , x n in its interior. The function l(x) captures the effect of the spiked eigenvalue and is given by [START_REF] Wang | The largest eigenvalue of real symmetric, Hermitian and Hermitian self-dual random matrix models with rank one external source, part i[END_REF][START_REF] Dharmawansa | Three problems related to the eigenvalues of complex non-central Wishart matrices with rank-1 mean[END_REF] (see also [START_REF] Onatski | Asymptotic power of sphericity tests for highdimensional data[END_REF][START_REF] Mo | Rank 1 real Wishart spiked model[END_REF])

l(z) = exp δ 1+δ z , for Model A 0 F 1 (m -n + 1, nνz), for Model B
where p F q (•) represents a hypergeometric function.

For Model C, it turns out that an analogous representation also exists. This is given by the following new result: Lemma 1. Under Model C, let x j ∈ (0, ∞), 1 ≤ j ≤ n denote the eigenvalues of F. Then, the joint density of f j = x j /(1 + x j ) ∈ (0, 1), 1 ≤ j ≤ n has the form

K n 2πı C 1 F 1 (m 1 + m 2 -n + 1, m 1 -n + 1, nνz) n j=1 f m 1 -n j (1 -f j ) m 2 -n z -f j 1≤j<k≤n (f k -f j ) 2 dz, (2)
where K n is a normalization constant, and the contour C encloses counter-clockwise f 1 , . . . , f n in its interior.

See Section 5.1 for the proof.

Based on (1) and (2), in the following we will compute the asymptotic distribution of linear spectral statistics for each of the three matrix models. In taking asymptotics, for Models A and B, we will be concerned with the following limits:

Assumption 1. m, n → ∞ such that m/n → c ≥ 1.
For Model C, we will be concerned with:

Assumption 2. m 1 , m 2 , n → ∞ such that m 1 /n → c 1 > 1 and m 2 /n → c 2 > 1.

Main Results

The two theorems below present the main contributions of the paper. In each case, x j ∈ (0, ∞), 1 ≤ j ≤ n will represent the eigenvalues of each associated matrix model.

a = (1 - √ c) 2 , b = (1 + √ c) 2 . ( 3
)
Under Assumption 1, for an analytic function f : U → C where U is an open subset of the complex plane which contains [a, b], we have

n k=1 f x k n L → N nµ + μ(z 0 ), σ 2 , (4)
where

µ = 1 2π b a f (x) (b -x)(x -a) x dx (5)
σ 2 = 1 2π 2 b a f (x) (b -x)(x -a) P b a f ′ (y) (b -y)(y -a)
xy dy dx [START_REF] Matthaiou | Sum rate analysis of zf receivers in distributed mimo systems[END_REF] with these terms independent of the spike. The spike-dependent term μ(z 0 ) admits

μ(z 0 ) = 1 2π b a f (x) (b -x)(x -a) (z 0 -a)(z 0 -b) z 0 -x -1 dx (7)
where

z 0 = (1+cδ)(1+δ) δ , for Model A (1+ν)(c+ν) ν , for Model B. (8)
The branch of the square root (z 0a)(z 0b) is chosen according to Remark 1 for Model A (see Section 5.2.1) and Remark 2 for Model B (see Section 5.2.2).

Theorem 2. Consider Model C. Define

a = c 1 (c 1 + c 2 -1) + c 2 -2 c 1 c 2 (c 1 + c 2 -1) (c 1 + c 2 ) 2 , b = c 1 (c 1 + c 2 -1) + c 2 + 2 c 1 c 2 (c 1 + c 2 -1) (c 1 + c 2 ) 2 . ( 9
)
Under Assumption 2, for an analytic function f : U → C where U is an open subset of the complex plane which contains [a, b], we have

n k=1 f (x k ) L → N nµ F + μF (z 0 ), σ 2 F ( 10
)
where

µ F = c 1 + c 2 2π b a f x 1 -x (b -x)(x -a) x(1 -x) dx ( 11
)
σ 2 F = 1 2π 2 b a f x 1-x (b -x)(x -a)   P b a f ′ y 1-y (b -y)(y -a)
xy dy   dx [START_REF] Chen | Coulumb fluid, Painlevé transcendents, and the information theory of MIMO systems[END_REF] with these terms independent of the spike. The spike-dependent term μF (z 0 ) admits

μF (z 0 ) = 1 2π b a f x 1-x (b -x)(x -a) (z 0 -a)(z 0 -b) z 0 -x -1 dx ( 13
)
where

z 0 = (1 + ν)(c 1 + ν) ν(c 1 + c 2 + ν) . ( 14
)
The branch of the square root (z 0a)(z 0b) is chosen according to Remark 3 (see Section 5.3).

These results show that, for all three models, the asymptotic contribution coming from the spiked eigenvalue contributes to a O(1) correction to the mean of the linear statistic. In the absence of a spike, it turns out that such O(1) terms disappear, which is consistent with prior results in [START_REF] Chen | On the linear statistics of Hermitian random matrices[END_REF][START_REF] Chen | Coulumb fluid, Painlevé transcendents, and the information theory of MIMO systems[END_REF]. In particular, we have: Corollary 1. If for Model A, δ = 0, or for Model B, ν = 0, then (4) reduces to

n k=1 f x k n L → N nµ, σ 2 .
Corollary 2. If for Model C, ν = 0, then (10) reduces to

n k=1 f (x k ) L → N nµ F , σ 2 F .
The proofs for all results in this section are given in Section 5.

Some Example Applications

In this section, to illustrate the utility of our main results, we consider a specific application of relevance for each of the three random matrix models. These applications are quite different; each involving a different linear statistic. For Model A, we will reproduce a known result, whilst for Models B and C we will present results which are new. These are simply illustrative examples and our general results may apply to a much broader range of problems.

Model A: Likelihood ratio test of Σ = I n

As an application of Theorem 1 for Model A, we consider the classical LRT that the population covariance matrix is the identity, under a rank-one spiked population alternative. We will recover an existing result from [START_REF] Wang | A note on the CLT of the LSS for sample covariance matrix from a spiked population model[END_REF] and [START_REF] Onatski | Asymptotic power of sphericity tests for highdimensional data[END_REF], which was derived by more complicated means. Specifically, consider the m samples y 1 , . . . , y m , drawn from a n-dimensional complex Gaussian distribution with covariance matrix Σ. We aim to test the hypothesis:

H 0 : Σ = I n .
This test has been studied extensively in classical settings (i.e., n fixed, m → ∞), first in detail in [START_REF] Mauchly | Significance test for sphericity of a normal n-variate distribution[END_REF]. Denoting the sample covariance by S m = m -1 m k=1 y k y † k , the LRT is based on the linear statistic (see [START_REF] Anderson | An Introduction to Multivariate Statistical Analysis, 3rd Edition[END_REF]Chapter 10])

L = tr(S m ) -ln(det S m ) -1.
Under H 0 , with n fixed, as m → ∞, mL is well known to follow a χ2 distribution. However, with high-dimensional data for which the dimension n is large and comparable to the sample size m, the χ 2 approximation is no longer valid (see [START_REF] Bai | Corrections to LRT on large-dimensional covariance matrix by RMT[END_REF]). In this case, a better approach is to use results based on the double-asymptotic given by Assumption 1. Such a study has been done first under H 0 and later under the spike alternative H 1 . More specifically, under H 0 , this was presented in [START_REF] Bai | Corrections to LRT on large-dimensional covariance matrix by RMT[END_REF] using a CLT framework established in [START_REF] Bai | CLT for linear spectral statistics of large-dimensional sample covariance matrices[END_REF]. Under H 1 : "Σ has a spiked covariance structure as in Model A", this problem was addressed only very recently in the independent works 2 , [START_REF] Wang | A note on the CLT of the LSS for sample covariance matrix from a spiked population model[END_REF] and [START_REF] Onatski | Asymptotic power of sphericity tests for highdimensional data[END_REF]. The result in [START_REF] Wang | A note on the CLT of the LSS for sample covariance matrix from a spiked population model[END_REF] was again based on the CLT framework of [START_REF] Bai | CLT for linear spectral statistics of large-dimensional sample covariance matrices[END_REF], with their derivation requiring the calculation of contour integrals. The same result was presented in [START_REF] Onatski | Asymptotic power of sphericity tests for highdimensional data[END_REF], in this case making use of sophisticated tools of contiguity and Le Cam's first and third lemmas [START_REF] Van Der | Asymptotic Statistics, Cambridge Series in Statistical and Probabilistic Mathematics[END_REF].

Here, we will adopt our general framework to recover the same result as [START_REF] Wang | A note on the CLT of the LSS for sample covariance matrix from a spiked population model[END_REF] and [START_REF] Onatski | Asymptotic power of sphericity tests for highdimensional data[END_REF] very efficiently, simply by calculating a few integrals. Under H 1 , as before we denote by 1 + δ the spiked eigenvalue of Σ. Since mS m ∼ CW n (m, Σ, 0 n×n ), we now apply Theorem 1 for the case of Model A to the function

f L (x) = x c -ln x c -1. Let x k , 1 ≤ k ≤ n, be the eigenvalues of mS m . Since the domain of definition of f L is (0, ∞), we assume that c > 1 to ensure a > 0 (see (3)). Then, under Assumption 1, L = n k=1 f L x k n L → N nµ L + μL , σ 2 L ,
where

µ L = 1 + (c -1) ln 1 -c -1 σ 2 L = -c -1 -ln 1 -c -1 with the spike-dependent term μL = δ -ln(1 + δ).
These results are in agreement with [START_REF] Wang | A note on the CLT of the LSS for sample covariance matrix from a spiked population model[END_REF] and [START_REF] Onatski | Asymptotic power of sphericity tests for highdimensional data[END_REF].

The expression for µ L was obtained by multiplying the numerator and the denominator of the integrand in ( 5

Model B: Capacity of MIMO Communication Systems with Line-of-Sight (LoS)

As an application of Theorem 1 for Model B, we consider the capacity of multipleantenna communication systems. In particular, consider a MIMO wireless communication system with n t transmit and n r receive antennas. The linear model relating the input (transmitted) signal vector x of size n t and output (received) signal vector y of size n r takes the form

y = Hx + n,
where n is a complex Gaussian vector of size n r , representing appropriately normalized receiver noise, with zero mean and covariance E(nn † ) = I nr . The n r ×n t matrix H represents the wireless fading coefficients (i.e, the channel gains between each pair of transmit and receive antennas), and this is assumed to be known to the receiver but not to the transmitter. We consider a communication scenario in which there is a direct LoS path between the transmitter and the receiver, with rich scattering in the communication environment. Under these assumptions, H is reasonably modeled as a complex Gaussian random matrix with independent entries and non-zero mean according to:

H = K K + 1 M + 1 K + 1 H w ,
where H w is an i.i.d. matrix with zero mean, unit variance complex Gaussian entries, M is deterministic and arbitrary, normalized such that tr(MM † ) = n r n t and K < ∞ is the so-called "Rician factor" between the unfaded (deterministic) and faded (random) components. For consistency with Model B, we will assume that

K = K 0 / max(n r , n t )
for fixed K 0 . Thus for large numbers of antennas, our results will formally apply for scenarios for which the Rician K factor is not too strong. Furthermore, with a direct LoS path, we make the natural assumption that M † M is of rank one, so that its sole non-null eigenvalue is n r n t . In the absence of any information about H, the transmitted signals are assumed to obey x ∼ CN 0 nt×1 , P nt I nt , where P is the total transmit power which is assumed to be spread equally across all antennas. Note that since the noise variance is normalized to unity, P also represents the signal to noise ratio (SNR).

In terms of performance evaluation, one of the most fundamental performance metrics is the "outage probability", which relates immediately to the distribution of the quantity

C = ln det I nr + P n t HH † (15)
which is an information-theoretic quantity reflecting the mutual information between the transmitted and received signals. This has been studied extensively for over a decade (see e.g., [START_REF] Chen | Coulumb fluid, Painlevé transcendents, and the information theory of MIMO systems[END_REF][START_REF] Li | On the distribution of MIMO mutual information: An in-depth Painlevé based characterization[END_REF][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] Chiani | On the capacity of spatially correlated MIMO Rayleigh-fading channels[END_REF][START_REF] Smith | Capacity of MIMO systems with semicorrelated flat fading[END_REF]) under various different assumptions; for example, assuming different distributions for the channel matrix.

For MIMO communication systems with LoS, the asymptotic distribution of ( 15) for large numbers of antennas has been studied in [START_REF] Kammoun | On the fluctuations of the mutual information for non centered MIMO channels: The non Gaussian case[END_REF][START_REF] Hachem | Deterministic equivalents for certain functionals of large random matrices[END_REF] via Steiltjes transform methods and in [START_REF] Moustakas | Random matrix theory of multi-antenna communications: The Rician channel[END_REF][START_REF] Taricco | Asymptotic mutual information statistics of separately correlated Rician fading MIMO channels[END_REF] via the replica method. Nevertheless, such results were not explicit: they were expressed in terms of solutions of fixed-point equations requiring numerical evaluation. Here, we will find an explicit expression for the asymptotic distribution of (15) which, to the best of our knowledge, is new.

Before presenting this expression, we also point out that various results have also been obtained for the mean and variance of the mutual information [START_REF] Johnstone | On the distribution of the largest eigenvalue in principal components analysis[END_REF] for finite numbers of antennas. Such formulas are rather complicated; e.g., involving determinants, confluent hypergeometric functions and Meijer-G functions [START_REF] Kang | Capacity of MIMO Rician channels[END_REF], infinite series of exponential integral functions [START_REF] Alfano | Mutual information and eigenvalue distribution of MIMO Ricean channels[END_REF], or multi-dimensional integrals [START_REF] Jayaweera | On the capacity of multiple-antenna systems in Rician fading[END_REF]; or they are derived under bounds or alternative asymptotics such as high or low SNRs (see [START_REF] Hansen | A geometrical investigation of the rank-1 Rician MIMO channel at high SNR[END_REF][START_REF] Mckay | General capacity bound for spatially correlated Rician MIMO channels[END_REF][START_REF] Cui | Generic procedure for tightly bounding the capacity of MIMO correlated Rician fading channels[END_REF][START_REF] Mckay | Improved lower general bound for spatially-correlated Rician MIMO capacity[END_REF][START_REF] Mckay | New properties of complex noncentral quadratic forms and bounds on MIMO mutual information[END_REF][START_REF] Jin | On the ergodic capacity of rank-1 Ricean-fading MIMO channels[END_REF] and references therein).

To place the linear statistic [START_REF] Johnstone | On the distribution of the largest eigenvalue in principal components analysis[END_REF] in the context of our framework (in accordance with Model B), it is convenient to first define m = max{n r , n t }, n = min{n r , n t }, and

W = (K + 1)HH † , n r < n t (K + 1)H † H, n r ≥ n t .
We see that W ∼ CW n (m, I n , Θ), with

Θ = KMM † , n r < n t KM † M, n r ≥ n t
having the sole non-null eigenvalue Knm = K 0 n. Thus, in accordance with Model B, we set ν = K 0 . We will apply Theorem 1 for the case of Model B with the function

f C (x) = ln 1 + x T ,
where we have defined

T = n t nP K 0 m + 1 . Let x k , 1 ≤ k ≤ n, be the eigenvalues of W. Then, under Assumption 1, we obtain C = n k=1 f C x k n L → N nµ C + μC , σ 2 C ( 16
)
where

µ C = 1 2 (a + b) ln √ T + a + √ T + b 2 - ( √ T + a - √ T + b) 2 2 - √ ab ln ( √ ab + (T + a)(T + b)) 2 -T 2 ( √ a + √ b) 2 -2 ln(T ) σ 2 C = 2 ln 1 2 T + a T + b 1 4 + 1 2 T + b T + a 1 4 with μC = 1 2 ln 2(T ν + (1 + ν)(c + ν)) 2 ν 2 (1 + c + T )A + 2cν(1 + c + T + A) + ν 2 (T 2 + 2T (1 + c) + 1 + c 2 ) + 2c 2 .
Here, a and b are defined as in (3), and A = (T + a)(T + b).

The expressions for µ C and σ 2 C were calculated in [START_REF] Chen | Coulumb fluid, Painlevé transcendents, and the information theory of MIMO systems[END_REF][START_REF] Kazakopoulos | Living at the edge: A large deviations approach to the outage MIMO capacity[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] Tulino | Asymptotic outage capacity of multiantenna channels[END_REF][START_REF] Hachem | A new approach for mutual information analysis of large dimensional multi-antenna channels[END_REF] (which considered the mutual information distribution for zero-mean H), whilst μC was obtained using (C.1), (C.14), and (C.17), replacing a, b and z 0 by their respective values and taking into account [START_REF] Telatar | Capacity of multi-antenna Gaussian channels[END_REF].

Figure 1 plots the density of the normalized mutual information C/n for various n and K. The crosses represent the simulated PDF and the solid curve is a Gaussian distribution with mean µ C + μC (z 0 )/n and variance σ 2 C /n 2 (16). The close fit of our Gaussian approximation is evident in all cases, even when n is not large.

High SNR Behavior

For practical channels with sufficient dynamics (e.g., high mobility), one is often interested in the expected mutual information E(C), rather than the entire distribution. For this quantity, when the SNR P is large our result yields

nµ C ∼ P →∞ n [ln(P ) -L ∞ (K 0 )] + o P (1) (17)
where

L ∞ (K 0 ) = 1 + (c -1) ln c -1 c + ln K 0 m + 1 - 1 n ln 1 + c -1 K 0
is the so-called "power offset", and o P (1) → 0 when P → ∞ . In [START_REF] Lozano | High-SNR power offset in multiantenna communication[END_REF], the authors consider an analogous scenario, but with a fixed Rician factor K. They give a finite n, m formula for the power offset L ∞ . For comparison purpose, we set K = K 0 /m in their result, which gives

nµ C ∼ P →∞ n [ln(P ) -L ∞ (K 0 )] + o P (1) (18)
where

L ∞ (K 0 ) = 1 + (c -1) ln c -1 c + ln K 0 m + 1 - K 0 m ln(2) 2 F 2 (1, 1; 2, m + 1; -nK 0 ).
To leading order in n, the two results match: the difference is only in the O(1) term. Our logarithm term serves as an approximation of the 2 F 2 term. The accuracy of our approximation is demonstrated in Figure 2. Here, the curve "Approx. (log term)" corresponds to [START_REF] Baik | Eigenvalues of large sample covariance matrices of spiked population models[END_REF], whilst the curve "Approx. ( 2 F 2 )" corresponds to the result (18) from [START_REF] Lozano | High-SNR power offset in multiantenna communication[END_REF]. Despite the simplicity of our formula [START_REF] Baik | Eigenvalues of large sample covariance matrices of spiked population models[END_REF], we see that it is very accurate, even for small values of m and n. As a further point of reference, the exact expected value of C in ( 15) is also shown, where this was obtained by numerical simulation.

Model C: High-Dimensional Multiple Sample Significance Test

As an application of Theorem 2 for Model C, we consider the multiple significance test problem. In particular, consider q Gaussian populations CN (u (j) , Σ) of dimension n, with 1 ≤ j ≤ q and for each population, we assume that we have a sample of size p j : y (j) 1 , . . . , y (j) p j . We aim to test the hypothesis

H 0 : u (1) = • • • = u (q) = 0 n×1 .
This test has been studied in classical settings (i.e., n fixed, p j → ∞), see [START_REF] Anderson | An Introduction to Multivariate Statistical Analysis, 3rd Edition[END_REF]. Define p = q j=1 p j . The likelihood ratio statistic can be written as

Λ = det (I n + F) -1 , (19)
with

F = Σ-1 BA B † , ( 20
) B = ȳ(1) -ȳ(q) , . . . , ȳ(q-1) -ȳ(q) Σ = j,k (y (j) k - ȳ(j) k )(y (j) k - ȳ(j) k ) † , where ȳ(j) = p -1 j p j k=1 y (j)
k is the empirical mean of each population, and

A =       p 1 (p-p 1 ) p -p 1 p 2 p • • • -p 1 p q-1 p -p 1 p 2 p p 2 (p-p 2 ) p • • • -p 2 p q-1 p . . . . . . . . . . . . -p 1 p q-1 p -p 2 p q-1 p • • • p q-1 (p-p q-1 ) p       .
As seen from [START_REF] Anderson | An Introduction to Multivariate Statistical Analysis, 3rd Edition[END_REF], Σ and BA B † are independent, and

Σ ∼ CW n (p -q, Σ, 0 (p-q)×(p-q) ), BA B † ∼ CW n (q -1, Σ, Σ -1 BAB † ), (21)
where

B = u (1)
u (q) , . . . , u (q-1)u (q) . Under H 0 , with n fixed, as p → ∞, -p ln Λ is well-known to follow a χ 2 distribution. However, with high-dimensional data for which the dimension n of the data is large and comparable to the sample size p, the χ 2 approximation is no longer valid (see [START_REF] Bai | Testing linear hypotheses in high-dimensional regressions[END_REF]). In this case, a better approach is to use results based on the double asymptotic given by Assumption 2 with m 1 = q -1 > n and m 2 = pq. Such a study has been done under H 0 in [START_REF] Bai | Testing linear hypotheses in high-dimensional regressions[END_REF]. More specifically, this was presented as a special case of a more general linear hypothesis test in regression analysis, using a CLT framework established in [START_REF] Zheng | Central limit theorems for linear spectral statistics of large dimensional F -matrices[END_REF].

Compared with H 0 , under alternative hypotheses, less is known. Here, we will find the distribution ofln Λ under the specific alternative

H 1 : u (1) = 0 and u (2) = • • • = u (q) = 0 n×1 . (22)
Whilst this scenario has been considered previously under classical settings (see e.g., [START_REF] Schatzoff | Comparative robustness of six tests in multivariate analysis of variance[END_REF][START_REF] Olson | Comparative robustness of six tests in multivariate analysis of variance[END_REF]), with high-dimensional data and under the double asymptotic given by Assumption 2, it has not. Thus, the result which we present in the following is new. This result will permit the calculation of the asymptotic power of this test under the alternative [START_REF] Rao Nadakuditi | Sample eigenvalue based detection of highdimensional signals in white noise using relatively few samples[END_REF].

Denote by u 1 , . . . , u n the elements of the vector u (1) . Under H 1 above, we find after some calculations that the non-centrality matrix Σ -1 BAB † in (21) has only one non-null eigenvalue, given by ν = ξ(p 1p 2 1 /p) n k=1 u 2 k , where ξ is the top-left entry of Σ -1 . We set m 1 = q -1, m 2 = pq and assume that q > n + 1. For consistency with Model B, we will assume that p 1 is fixed, and either n k=1 u 2 k = K 1 n and ξ = K 2 , or n k=1 u 2 k = K 1 and ξ = K 2 n, where K 1 , K 2 > 0. Under these conditions, ν is constant under Assumption 2. Thus, F in (20) conforms to Model C. We can derive an explicit asymptotic characterization of the statisticln Λ (19) by applying Theorem 2 with the function

f R (x) = ln(1 + x). Let x k , 1 ≤ k ≤ n, be the eigenvalues of F. Then, under Assumption 2, -ln Λ = n k=1 f R (x k ) L → N nµ R + μR , σ 2 R , (23)
where

µ R = -(c 1 + c 2 ) ln √ 1 -a + √ 1 -b 2 - √ ab 2 ln 1 -( √ ab -(1 -a)(1 -b)) 2 ( √ a + √ b) 2 + (1 -a)(1 -b) ln 1 2 √ 1 -a + 1 2 √ 1 -b σ 2 R = ln ( √ 1 -a + √ 1 -b) 2 4 (1 -a)(1 -b) with μR = ln 1 + ν c 1 + c 2 . ( 24
)
Here, a and b are defined as in [START_REF] Diaconis | Linear functional of eigenvalues of random matrices[END_REF].

Note that the expressions for nµ R and σ 2 R are in agreement with previous results for the case of H 0 considered in [START_REF] Bai | Testing linear hypotheses in high-dimensional regressions[END_REF], for which u (1) is zero. Thus, the key difference under the H 1 scenario considered here is the non-zero term [START_REF] Couillet | Random Matrix Methods for Wireless Communications[END_REF], which serves as a perturbation to the asymptotic mean of the statistic.

In deriving the above results, the expression for µ R was obtained by multiplying the numerator and the denominator of the integrand in ( 11) by (bx)(xa), applying a partial fraction decomposition, and integrating using the identities (C.7)-(C.9). The expression for σ 2 R was obtained using (C.7), (C.9) and (C.12). The term μR was evaluated using (C.7) and (C.16), replacing a, b and z 0 by their respective values, and taking into account [START_REF] Cui | Generic procedure for tightly bounding the capacity of MIMO correlated Rician fading channels[END_REF].

Figure 3a plots the density of the linear spectral statisticln Λ for various n. The crosses represent the simulated PDF and the solid curve is a Gaussian distribution with mean nµ R + μR and variance σ 2 R [START_REF] Torun | Portfolio risk in multiple frequencies[END_REF]. The close fit of our Gaussian approximation is evident in all cases, even when n is not large.

Based on the above results, we may also compute the asymptotic power of the statistical test, which represents the probability that we reject H 0 when under H 1 . From standard hypothesis testing theory (see e.g. [START_REF] Walpole | Probability and Statistics for Engineers and Scientists[END_REF]), using4 (23), we get

β(α, ν, c 1 , c 2 ) = 1 -Φ Φ -1 (1 -α) - 1 σ R ln 1 + ν c 1 + c 2 , ( 25
)
where α is a parameter specifying the nominal level of the test, i.e., reflecting the probability of rejecting H 0 under H 0 , whilst Φ is the cumulative distribution function of a standard Gaussian and Φ -1 its inverse. The result is shown in Figure 3b for a nominal level α = 0.05. Results are shown for q = 21 populations, aggregate sample size of p = 51, with the first population having sample size p 1 = 5. The population covariance matrix is Σ = I n . The power is plotted as a function of the squared norm n k=1 u 2 k . Simulations are also shown for further comparison. Note that for the simulation results, the same statistical test is assumed as indicated above 5 , but now the exact power of this test is computed via Monte Carlo simulations. The close fit of our power approximation (25) is evident.

Derivations of Main Results

This section compiles the proofs of the key technical results in the paper.

Proof of Lemma 1

We adapt the derivation of [START_REF] Wang | The largest eigenvalue of real symmetric, Hermitian and Hermitian self-dual random matrix models with rank one external source, part i[END_REF][START_REF] Dharmawansa | Three problems related to the eigenvalues of complex non-central Wishart matrices with rank-1 mean[END_REF], which derived the joint eigenvalue density (1) for Model B. The joint eigenvalue density for Model C admits [START_REF] James | Distributions of matrix variates and latent roots derived from normal samples[END_REF]

p F (x 1 , . . . , x n ) = K n,m 1 ,m 2 e -tr(MM † ) 1 F1 (m 1 + m 2 ; m 1 ; Ω, Φ) × n j=1 x m 1 -n j (1 + x j ) m 1 +m 2 1≤j<k≤n (x k -x j ) 2 (26)
where

K n,m 1 ,m 2 is a constant, Φ = diag(f 1 , . . . , f n ), f j = x j /(1 + x j ), Ω = diag(ω 1 , . . . , ω n ),
ω j is the jth eigenvalue of Θ and 1 F1 (•; •; •, •) denotes the confluent hypergeometric function of two matrix arguments.

The crux of the proof lies in an alternative contour-integral representation which we present for the 1 F1 function under the spiked model. To this end, we start with the traditional expansion [START_REF] James | Distributions of matrix variates and latent roots derived from normal samples[END_REF][START_REF] Takemura | Zonal Polynomials[END_REF],

1 F1 (m 1 + m 2 ; m 1 ; Ω, Φ) = ∞ k=0 1 k! κ [m 1 + m 2 ] κ [m 1 ] κ C κ (Ω)C κ (Φ) C κ (I n ) , ( 27
)
where

C κ (•) is a complex zonal polynomial, whilst κ = (k 1 , . . . , k n ) with k j ∈ N is a partition of k such that k 1 ≥ • • • ≥ k n ≥ 0 and n j=1 k j = k. Moreover, [ℓ] κ = ℓ j=1 (ℓ -j + 1) k j , where (ℓ) k = Γ(ℓ + k)/Γ(ℓ)
denotes the Pochhammer symbol. In our case, the only nonnull eigenvalue of Ω is the spiked eigenvalue nν. Thus, from the definition of C κ (•) (see [START_REF] James | Distributions of matrix variates and latent roots derived from normal samples[END_REF][START_REF] Takemura | Zonal Polynomials[END_REF]), it follows that C κ (Ω) = 0 for all partitions of k having more than one non-zero part. Therefore, only partitions of the form (k, 0, . . . , 0), which we denote by k, contribute to the summation. Furthermore,

C k (I n ) = k-1 j=0 n+j 1+j . Consequently, (27) reduces to 1 F1 (m 1 + m 2 ; m 1 ; Ω, Φ) = ∞ k=0 1 k! (m 1 + m 2 ) k (m 1 ) k k-1 j=0 1 + j n + j C k (Φ)(nν) k (28)
which is seen as a power series expansion in ν. Following [START_REF] Wang | The largest eigenvalue of real symmetric, Hermitian and Hermitian self-dual random matrix models with rank one external source, part i[END_REF], and recalling the definition of Φ above, we also have

1 k! k-1 j=0 1 + j C k (Φ) = 1 2πı 0 n j=1 1 1 -zf j dz z k+1
where the contour is taken to be a small circle around 0 taken counter-clockwise with 1/f j , 1 ≤ j ≤ n being exterior to the contour. Using this result in [START_REF] Bai | CLT for linear spectral statistics of large-dimensional sample covariance matrices[END_REF], upon exchanging the summation and integral by applying the dominated convergence theorem, we obtain

1 F1 (m 1 + m 2 ; m 2 ; Ω, Φ) = 1 2πı 0 n j=1 1 1 -zf j ∞ k=0 (m 1 + m 2 ) k (m 1 ) k (n) k (nν) k z k+1 dz.
Defining N = n -1, we can further write

1 F1 (m 1 + m 2 ; m 1 ; Ω, Φ) = N !(m 1 -1)! (m 1 + m 2 -1)! 1 2πı 0 n j=1 1 1 -zf j ∞ k=N Γ(m 1 + m 2 + k -N ) Γ(m 1 + k -N )k! (nν) k-N z k-N +1 dz = ψ 2πı 0 n j=1 1 1 -zf j ∞ k=0 (m 1 + m 2 -N ) k k!(m 1 -N ) k (nν) k-N z k-N +1 - N -1 k=0 (m 1 + m 2 -N ) k k!(m 1 -N ) k (nν) k-N z k-N +1 dz
where for notational convenience we have defined

ψ = N !(m 1 -1)!(m 1 + m 2 -n)! (m 1 + m 2 -1)!(m 1 -n)! .
Since the integrand is an analytic function, the second sum is zero. Further recognizing the first sum as a scalar 1 F 1 hypergeometric function (up to a scaling), we may then write

1 F1 (m 1 + m 2 ; m 1 ; Ω, Φ) = ψ (nν) N 1 2πı 0 n j=1 1 1 -zf j z N -1 1 F 1 m 1 + m 2 -N, m 1 -N, nν z dz,
which upon applying the transformation z → 1/z yields

1 F1 (m 1 + m 2 ; m 2 ; Ω, Φ) = ψ (nν) N 1 2πı C n j=1 1 z -f j 1 F 1 (m 1 + m 2 -N, m 1 -N, nνz) dz,
where the contour C now encloses counter-clockwise all the f j , 1 ≤ j ≤ n. Substituting this expression into [START_REF] Couillet | Fluctuations of spiked random matrix models and failure diagnosis in sensor networks[END_REF] and applying the change of variable x k = f k 1-f k , we obtain the desired result.

Proof of Theorem 1 (Models A and B)

Our strategy is to evaluate the moment generating function (MGF), which is

M(λ) = E e λ n k=1 f (x k /n) .
Using [START_REF] Ledoit | Honey, I shrunk the sample covariance matrix[END_REF], upon applying the transformations x j → nx j and z → nz, we obtain

M(λ) = K n [l] 2πı n m+1 C l(nz)Z n (λ, z) dz ( 29
)
where

Z n (λ, z) = R n + 1≤j<k≤n (x k -x j ) 2 n j=1
x m-n j e -nx j zx j e λf (x j ) dx j and the contour C encloses now counter-clockwise all the scaled eigenvalues x 1 /n, . . . , x n /n in its interior.

It will be convenient to rewrite Z n (λ, z) in the equivalent form:

Z n (λ, z) = R n + e -Φ(x 1 ,...,xn)-n k=1 g(x k ) n k=1 dx k (30)
where

g(x) = g(x, z) = -λf (x) + ln(z -x), (31) with Φ
(x 1 , . . . , x n ) = -2 1≤j<k≤n ln |x j -x k | + n n j=1 v 0 (x j )
where we have defined

v 0 (x) = x - m n -1 ln x.
Setting g(x) = 0 in [START_REF] Wang | On the sphericity test with large-dimensional observations[END_REF], we also introduce

Z n = R n + e -Φ(x 1 ,...,xn) n k=1 dx k ,
which is simply a constant.

With this formulation, the results from [START_REF] Chen | On the linear statistics of Hermitian random matrices[END_REF] and also [START_REF] Chen | Coulumb fluid, Painlevé transcendents, and the information theory of MIMO systems[END_REF], derived based on the Coulomb fluid method, now immediately suggest that as n → ∞ with m/n → c,

Z n (λ, z) ≈ Z n e -S 1 (z) 2 -S 2 (z) (32)
where

S 1 (z) = b a g(x, z)̺(x, z) dx (33)
S 2 (z) = n b a g(x, z)σ 0 (x) dx. (34)
Here a = (1 -√ c) 2 and b = (1 + √ c) 2 , as defined in the theorem statement, whilst

σ0 (x) = 1 2π (b -x)(x -a) x , x ∈ [a, b] (35)
which is the Marčenko-Pastur law (see [START_REF] Dyson | An Ising ferromagnet with discontinuous long-range order[END_REF][START_REF] Marčenko | Distribution of eigenvalues in certain sets of random matrices[END_REF]). Also,

̺(x, z) = -λρ 1 (x) + ρ 2 (x, z) (36)
where

ρ 1 (x) = 1 2π 2 (b -x)(x -a) P b a (b -y)(y -a) y -x f ′ (y) dy and ρ 2 (x, z) = 1 2π 2 (b -x)(x -a) P b a (b -y)(y -a) y -x 1 y -z dy, x ∈ [a, b].
Multiplying the numerator and the denominator of the integrand by (by)(ya), applying a partial fraction decomposition and integrating using the identities (C.3), (C.10) and (C.13), we obtain

ρ 2 (x, z) = 1 2π (b -x)(x -a) (z -a)(z -b) z -x -1 . (37)
Consider S 1 . Plugging [START_REF] Chen | Thermodynamic relations of the Hermitian matrix ensembles[END_REF] along with ( 31) into (33) yields a quadratic in λ,

S 1 (z) = -λ 2 σ 2 -2λμ(z) -A 1 (z) (38)
where σ 2 takes the form ( 6), the linear coefficient μ(•) takes the form ( 7) since (see Appendix B for details)

μ(z) = 1 2 b a [f (x)ρ 2 (x, z) + ln(z -x)ρ 1 (x)] dx = b a f (x)ρ 2 (x, z) dx, (39)
whilst the constant term is

A 1 (z) = - b a ln(z -x)ρ 2 (x, z) dx. (40)
Note that this last term is independent of the linear statistic f (•) and will not contribute to either the asymptotic mean or variance. Now consider S 2 . Plugging ( 31) and ( 35) into [START_REF] Chen | Asymptotic level spacing of the Laguerre ensemble: A Coulomb fluid approach[END_REF] gives

S 2 (z) = -n (λµ + A 2 (z)) (41)
where µ takes the form (5), whilst

A 2 (z) = - 1 2π b a ln(z -x) (b -x)(x -a) x dx (42)
is a constant which will contribute to the asymptotic mean in the sequel.

Combining [START_REF] Van Der | Asymptotic Statistics, Cambridge Series in Statistical and Probabilistic Mathematics[END_REF] together with [START_REF] Simon | Crossover from conserving to lossy transport in circular random-matrix ensembles[END_REF] and [START_REF] Dean | Extreme value statistics of eigenvalues of Gaussian random matrices[END_REF], we obtain

Z n (λ, z) ≈ Z n e λ 2 σ 2 2 +λ[nµ+μ(z)]+ A 1 (z) 2 +nA 2 (z) . (43)
Substituting ( 43) into (29), we obtain for large n

M(λ) ∝ I(λ)e λ 2 σ 2 2 +λnµ (44)
with

I(λ) = C l(nz)e λμ(z)+ A 1 (z) 2 +nA 2 (z) dz. ( 45
)
The remaining challenge is to deal with this contour integral, seeking a solution for large n, which will be addressed using the Laplace approximation (or saddlepoint) method. We will consider Model A and Model B in turn.

Saddlepoint Evaluation for Model A

In this case, l(nz) = exp n δ 1+δ z . Plugging this into (45), the Laplace approximation yields (see [START_REF] Olver | Asymptotics and Special Functions[END_REF]Chapter 4] or [START_REF] Bleistein | Asymptotic Expansions of Integrals[END_REF]Chapter 7] for more details):

I(λ) = C e -np(z) q(z) dz ≈ exp(-np(z 0 )) 2π n q(z 0 ) p ′′ (z 0 ) (46)
for which

p(z) = - δ 1 + δ z + A 2 (z) (47)
q(z) = exp λμ(z) + A 1 (z) 2 (48)
and where z 0 is the saddlepoint, which is the solution to

p ′ (z 0 ) = 0.
The final task is to evaluate z 0 . (Note that a similar saddlepoint problem was addressed in [START_REF] Onatski | Asymptotic power of sphericity tests for highdimensional data[END_REF]; we follow the same approach.) Substituting ( 42) into [START_REF] Mo | Rank 1 real Wishart spiked model[END_REF] and taking the derivative with respect to z, we find that z 0 must satisfy

δ 1 + δ + A ′ 2 (z 0 ) = 0, (49)
where

A ′ 2 (z 0 ) = 1 2π b a (b -x)(x -a) x(x -z 0 ) dx (50) = -z 0 + c -1 + (z 0 -a)(z 0 -b) 2z 0 (51) for z 0 / ∈ [a, b] ∪ {0}.
The second equality is obtained by multiplying the numerator and the denominator of the integrand in ( 50) by (bx)(xa), applying a partial fraction decomposition and integrating using the identities (C.2), (C.3) and (C.13). (Note that [START_REF] Mauchly | Significance test for sphericity of a normal n-variate distribution[END_REF] is related to the "usual" Stieltjes transform of the Marčenko-Pastur law (see e.g., [START_REF] Bai | Convergence rate of expected spectral distributions of large random matrices. II. Sample covariance matrices[END_REF]) via the changes of variable x → cx.) The solution z 0 to (49) is

z 0 = (1 + cδ)(1 + δ) δ .
Remark 1. In order to have a solution to (49) outside [a, b], we have to take the following specific branches for the square root in (51):

• When 0 < δ ≤ 1/ √ c
, the branch is chosen so that the signs of the real and imaginary part of (z 0a)(z 0b) match those of z 0c -1;

• When δ > 1/ √ c, the signs are chosen to be opposite.

The square root in both cases then evaluates to the common form:

(z 0 -a)(z 0 -b) = 1 -cδ 2 δ . (52)
When substituting for q(z 0 ) in ( 46) using ( 48), [START_REF] Vivo | Distributions of conductance and shot noise and associated phase transitions[END_REF], and (37), we again encounter the same square root (this is the one indicated in the theorem statement). For this, the same branch should be taken as indicated above.

Saddlepoint Evaluation for Model B

In this case, l(nz) = 0 F 1 (mn + 1, n 2 νz). In order to apply a method similar to Model A, we require an "exponential type" representation or approximation for n large for the hypergeometric function 0 F 1 . To this end, note that 0 F 1 can be written in terms of a modified Bessel function I α (z) [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF]

0 F 1 (α + 1, z) = Γ(α + 1)z -α 2 I α (2 √ z).
Thus we have

l(nz) = Γ(m -n + 1) n m-n (νz) m-n 2 I m-n 2n √ νz .
Moreover, when the parameter α → ∞ and | arg(z)| < π 2ε, for ε > 0, we have the asymptotic expansion [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF][START_REF] Olver | Asymptotics and Special Functions[END_REF]

I α (αz) ≈ e α √ 1+z 2 +α ln z 1+ √ 1+z 2 √ 2πα(1 + z 2 ) 1/4 1 + 3z 2 -2 24α(1 + z 2 ) 3 2 + • • • .
Since mn ∼ n(c -1) → ∞ when n → ∞, we can use the above result together with the Stirling approximation [START_REF] Tweddle | James Stirling's Methodus differentialis[END_REF] to obtain that, when n → ∞,

l(nz) ≈ (c -1) c-1 2
e n(c-1) (νz)

n(c-1) 2 exp n (c -1) 2 + 4νz + n(c -1) ln 2 √ νz c-1+ √ (c-1) 2 +4νz (c -1) 2 + 4νz for | arg(2 √ νz/(c -1))| < π 2 -ε, ε > 0.
Using this approximation in [START_REF] Chen | Random matrix models, double-time Painlevé equations, and wireless relaying[END_REF] and keeping only the terms dependent on z, the resulting contour integral takes the Laplace form [START_REF] Wang | The largest eigenvalue of real symmetric, Hermitian and Hermitian self-dual random matrix models with rank one external source, part i[END_REF] with p(z) and q(z) as follows:

p(z) = - (c -1) 2 + 4νz + (1 -c) ln c -1 + (c -1) 2 + 4νz + A 2 (z) (53)
q(z) = exp λμ(z) + A 1 (z) 2 + (c -1) ln(2 √ ν) -ln((c -1) 2 + 4νz) . (54)
Substituting ( 42) into [START_REF] Onatski | Signal detection in high dimension: The multispiked case[END_REF] and taking the derivative with respect to z, we find that the saddlepoint z 0 must satisfy 2ν

c -1 + (c -1) 2 + 4νz 0 + A ′ 2 (z 0 ) = 0, (55)
where A ′ 2 (z 0 ) is defined by [START_REF] Mauchly | Significance test for sphericity of a normal n-variate distribution[END_REF]. The solution z 0 to (55) is

z 0 = (1 + ν)(c + ν) ν .
Remark 2. In order to have a solution to (55) outside [a, b], we have to take the following specific branches for the square root in (51):

• When 0 < ν ≤ √ c, the branch is chosen so that the signs of the real and imaginary part of (z 0a)(z 0b) match those of z 0c -1;

• When ν > √ c, the signs are chosen to be opposite.

The square root in both cases then evaluates to the common form:

(z 0 -a)(z 0 -b) = c ν -ν. ( 56
)
As described previously, substituting for q(z 0 ) in ( 46) using ( 54), [START_REF] Vivo | Distributions of conductance and shot noise and associated phase transitions[END_REF], and (37) reveals the same square root (indicated in the theorem statement). For this, the same branch should be taken as indicated above.

Completing the Proof for Both Models

Finally, plugging [START_REF] Mo | Rank 1 real Wishart spiked model[END_REF] and [START_REF] Dharmawansa | Three problems related to the eigenvalues of complex non-central Wishart matrices with rank-1 mean[END_REF] (or ( 53) and ( 54)) into [START_REF] Wang | The largest eigenvalue of real symmetric, Hermitian and Hermitian self-dual random matrix models with rank one external source, part i[END_REF], we can rewrite [START_REF] Li | On the distribution of MIMO mutual information: An in-depth Painlevé based characterization[END_REF] as

M(λ) ∝ exp λ 2 σ 2 2 + λ (nµ + μ(z 0 )) + r(z 0 )
as n → ∞, where the function r(z 0 ) does not depend on λ. This is recognized as the MGF of a Gaussian distribution with mean nµ + μ(z 0 ) and variance σ2 .

Proof of Theorem 2 (Model C)

The proof is similar to Theorem 1. We will evaluate the MGF

M(λ) = E e λ n k=1 f (x k ) .
Using ( 2) we obtain

M(λ) = K n 2πı C l(z)Z n (λ, z) dz (57)
where

Z n (λ, z) = (0,1) n 1≤j<k≤n (f k -f j ) 2 n j=1 f m 1 -n j (1 -f j ) m 2 -n z -f j e λf f j 1-f j df j and l(z) = 1 F 1 (m 1 + m 2 -n + 1, m 1 -n + 1, nνz) .
Following the derivation of Theorem 1, in this case we obtain

v 0 (x) = (1 -c 1 ) ln x + (1 -c 2 ) ln(1 -x) σ0 (x) = c 1 + c 2 2π (b -x)(x -a) x(1 -x) , x ∈ [a, b]
with a and b defined as in the theorem statement (see [START_REF] Chen | Coulumb fluid, Painlevé transcendents, and the information theory of MIMO systems[END_REF] for more details, which considered the non-spike scenario with a specific linear statistic). Once again, using the result of [START_REF] Chen | On the linear statistics of Hermitian random matrices[END_REF] we have, as n → ∞ such that m 1 /n → c 1 and m 2 /n → c 2 ,

Z n (λ, z) ≈ Z n e -S 1 (z)
where σ 2 F takes the form [START_REF] Chen | Coulumb fluid, Painlevé transcendents, and the information theory of MIMO systems[END_REF], the linear coefficient μF (•) takes the form [START_REF] Bai | Testing linear hypotheses in high-dimensional regressions[END_REF], whilst the constant term is

A F,1 (z) = - b a ln(z -x)ρ 2 (x, z) dx, (59)
where

ρ 2 (x, z) = 1 2π (b -x)(x -a) (z -a)(z -b) z -x -1 . ( 60
)
This constant term A F,1 is independent of the linear statistic f (•) and will not contribute to either the asymptotic mean or variance. Now consider S 2 . We have

S 2 (z) = -n (λµ F -A F,2 (z))
where µ F takes the form [START_REF] Bai | Corrections to LRT on large-dimensional covariance matrix by RMT[END_REF], whilst

A F,2 (z) = - c 1 + c 2 2π b a ln(z -x) (b -x)(x -a) x(1 -x) dx.
Substituting [START_REF] Smith | Capacity of MIMO systems with semicorrelated flat fading[END_REF] into [START_REF] Chiani | On the capacity of spatially correlated MIMO Rayleigh-fading channels[END_REF] we obtain that, as n → ∞ with m 1 /n → c 1 and m 2 /n → c 2 ,

M(λ) ∝ I(λ)e λ 2 σ 2 F 2 +λnµ F (61)
with

I(λ) = C l(z)e λμ F (z)+ A F,1 (z) 2 +nA F,2 (z) dz.
As before, to deal with the contour integral, we seek a saddlepoint approximation for large n. Using the asymptotic approximation of l(z) given in Lemma 2 with u =

m 1 /n + m 2 /n -1, v = m 1 /n -1 and γ = ν (see Appendix A) we have, as n → ∞ I(λ) ∝ C e nνzt(z) t(z) n(c 1 +c 2 -1)+1 (t(z) -1) -nc 2 (t(z) -1) 2 (1 -c 1 ) + c 2 (2t(z) -1) e λμ F (z)+ A F,1 (z) 2 +nA F,2 (z) dz, where t(z) = νz + 1 -c 1 + (c 1 -1 -νz) 2 -4νz(1 -c 1 -c 2 ) 2νz .
The Laplace approximation yields

I(λ) = C e -np(z) q(z) dz ≈ exp(-np(z 0 )) 2π n q(z 0 ) p ′′ (z 0 ) (62)
for which

p(z) = -(νzt(z) + (c 1 + c 2 -1) ln(t(z)) -c 2 ln(t(z) -1) + A F,2 (z)) (63)
q(z) = exp λμ F (z) + A F,1 (z) 2 (t(z) -1) 2 (1 -c 1 ) + c 2 (2t(z) -1) (64)
and where z 0 is the saddlepoint, which is the solution to

p ′ (z 0 ) = 0.
This satisfies

ν(t(z 0 ) + t ′ (z 0 )z 0 ) + (c 1 + c 2 -1) t ′ (z 0 ) t(z 0 ) -c 2 t ′ (z 0 ) t(z 0 ) -1 + A ′ F,2 (z 0 ) = 0, (65)
where

A ′ F,2 (z 0 ) = - c 1 + c 2 2π b a (b -x)(x -a) x(1 -x)(z 0 -x) dx (66)
= c 1 + c 2 2 √ ab + (z 0 -a)(z 0 -b) -z 0 (1 -a)(1 -b) + √ ab z 0 (1 -z 0 ) (67)
for z 0 / ∈ [a, b] ∪ {0, 1}. The second equality is obtained by multiplying the numerator and the denominator of the integrand in (66) by (bx)(xa), applying a partial fraction decomposition and integrating using the identities (C.2) and (C.13). The solution z 0 to (67) is

z 0 = (1 + ν)(c 1 + ν) ν(c 1 + c 2 + ν) .
Remark 3. In order to have a solution to (65) outside [a, b], we have to take the following specific branches for the square root (z 0a)(z 0b) in (67):

• When 0 < ν ≤ c = c 1 + √ c 1 c 2 (c 1 +c 2 -1) c 2 -1
, the branch is chosen so that the signs of the real and imaginary parts match those of 1c 1 + (c 1 + c 2 )z 0 ;

• When ν > c, the signs are chosen to be opposite.

The square root in both cases then evaluates to the common form:

(z 0 -a)(z 0 -b) = c 1 (c 1 + c 2 ) + 2c 1 ν -(c 2 -1)ν 2 ν(c 1 + c 2 + ν)(c 1 + c 2 ) . ( 68
)
As described previously, substituting for q(z 0 ) in ( 46) using ( 64), [START_REF] Kammoun | On the fluctuations of the mutual information for non centered MIMO channels: The non Gaussian case[END_REF], and (60) reveals the same square root (indicated in the theorem statement). For this, the same branch should be taken as indicated above.

where b a f (x)ρ 2 (x, z)dx = 1 2π b a f (x) (b -x)(x -a) (z -a)(z -b) z -x -1 dx (B.1) and b a ln(z -x)ρ 1 (x) dx = 1 2π 2 b a ln(z -x) (b -x)(x -a) P b a f ′ (y) (b -y)(y -a) y -x dy dx. (B.2)
This identity is not straightforward and appears difficult to show directly using the above expressions. Thus, here we adopt an approach based on first showing that the derivative with respect to z of (B.1) and (B.2) are equal. First considering (B.1), we have Now, note that with z such that ℑz = 0, as ℜz → ∞,

d dz b a f (x)ρ 2 (x, z) dx = 1 2π (z -a)(z -b) b a f (x) ((a + b)(x + z) -2ab) 2(z -x) 2 (b -x)(x -
(z -a)(z -b) z -x -1 = x -a+b 2 z + O 1 z 2 .
Plugging this expression into (B.1) gives

b a f (x)ρ 2 (x, z)dx = 1 z b a f (x) (b -x)(x -a) x - a + b 2 dx + O 1 z 2
which tends to zero as ℜz → ∞. Furthermore, with ln(zx) = ln(z) - Here, the first principal value integral is zero by (C.10), whilst the remaining terms tends to zero when z is such that ℜz → ∞ and ℑz = 0.

Consequently, taking z such that ℜz → ∞ and ℑz = 0 in (B.4), we find that the constant term is zero, thus proving the result. are valid for ℜt > b, while in all cases we assume 0 < a < b. The first integral is given by (C.13), whereas the double integral is The first term on the right-hand side above is obtained using (C.13), whereas the second

Theorem 1 .

 1 Consider Models A and B. Define

  ) by (bx)(xa), applying a partial fraction decomposition, then integrating using the identities (C.1)-(C.6) along with [54, Eq. 2.264, 3.], and finally replacing a and b by their respective values. The expression for σ 2 L was obtained by using the integral identities (C.11) and (C.15), along with those indicated above. The spikedependent term μL was obtained by using (C.1)-(C.3) and (C.17), replacing a, b and z 0 by their respective values, and taking into account (52).

  m = 2n, c = 2, K 0 = 5.

  m = 32, n = 16, c = 2.

Figure 1 :

 1 Figure 1: PDF of C/n. Results shown for 3 P = 5 dB.

  m = 6, n = 4, K 0 = 30 (K = 5).

  m = 32, n = 16, K 0 = 90 (K = 5).

Figure 2 :

 2 Figure 2: Plot of E(C) as a function of the SNR, with three high-SNR approximations.

  p = 1 + 2n, q = 1 + 5n, p 1 = 1, p = 51, q = 21, p 1 = 5, n = 10.

Figure 3 :

 3 Figure 3: (a) PDF ofln Λ and (b) power function.

  y)(zx) (bx)(xa) dy = b a (by)(ya) 2π 2 (zy) f ′ (y) P b a (zx) -1 + (xy) -1 (bx)(xa)dx dy, which, after applying the identities (C.10) and (C.13), yieldsd dz b a ln(zx)ρ 1 (x) dx = 1 2π (za)(zb) b a (by)(ya) yz f ′ (y) dy.Application of integration by parts to the last integral givesd dz b a ln(zx)ρ 1 (x) dx = 1 2π (za)(zb) b a f (y) ((a + b)(y + z) -2ab) 2(zy) 2 (by)(ya)dy, which is the same as (B.3). So we have proved that b a ln(zx)ρ 1 (x) dx = b a f (x)ρ 2 (x, z)dx + Constant. (B.4)

  the expression in (B.2) becomes, upon interchanging the integrals, y) (by)(ya) P b a ln(z) dx(yx) (bx)(xa) x) (bx)(xa)dy.

2 ( 4 1 ( 1 ( 1 - 1 (

 241111 t)(bx)(xa) = π (t + a)(t + b) + t) (bx)(xa)x dx = π √ ab ln ( √ ab + (t + a)(t + b)) 2t + (a + t)(b + t)) + a + b yx) (bx)(xa) y)(ya) (1y)(xy) dy = π (1a)(1b) t) (bx)(xa) dx = -π (ta)(tb) x)(xa) yx dx = π y -a + b 2 (C.15) Moreover, for z ∈ C, b a ln(1x) (xz) (bx)(xa) dx = π A ln 2A + 2zab 2A (1a)(1b) + z(2ab)ab + 2ab (C.16) b a ln(x + t) dx (xz) (bx)(xa) = π A ln 2A + (a + b -2z)(t + z) + 2A A 2 + (a + b -2z)(t + z) + (t + z) 2 (t + z) 2 (a + b -2z + 2A) (C.17) with A = (za)(zb).Equations (C.1)-(C.12) are given in[START_REF] Chen | Coulumb fluid, Painlevé transcendents, and the information theory of MIMO systems[END_REF], whilst (C.13) is a slight modification of (C.2), and (C.14) follows using (C.2) and (C.3). The expression (C.15) follows upon multiplying the numerator and the denominator of the integrand by (bx)(xa), applying a partial fraction decomposition, then invoking (C.3), (C.4) and (C.10). For (C.16) and (C.17), the derivations are more involved, and we describe each in turn. For (C.16), we use the parametrization ln(1x) = fraction decomposition x(x -1/λ)(xz) = 1/λ (x -1/λ)(1/λz) -z (xz)(1/λz) to arrive at b a ln(1x) dx (xz) (bx)(xa) λa)(1λb) + z (za)(zb)dλ 1λz . The last equation was obtained by invoking (C.2) and (C.13). From a further change of variable x = 1λz, we have b a ln(1x)(xz) (bx)(xa) za + ax)(zb + bx) 1z) + ln 2A (za + ax)(zb + bx) + 2A 2 + x(z(a + b) -2ab) ab 2A (1a)(1b) + z(2ab)ab + 2ab .Now consider (C.17). In this case, we use the relation ln(x + t) = ln t + + t) dx (xz) (bx)(xa)

  z + a)(xz + b) [setting x = t/y + z] = π ∞ t+z dx x -(bz)(za) + (b + a -2z)x + x 2 = π A ln 2A + (a + b -2z)(t + z) + 2A A 2 + (a + b -2z)(t + z) + (t + z) 2 (t + z)(a + b -2z + 2A) ,using the first identity in [54, Eq. 2.266]. Combining the previous calculations, we get the result.

The paper[START_REF] Bai | CLT for linear spectral statistics of large-dimensional sample covariance matrices[END_REF] considered a more general model than the spiked model. Therein, CLT results were presented for linear spectral statistics, with the key quantities involving solutions to implicit equations.

We point out that[START_REF] Wang | A note on the CLT of the LSS for sample covariance matrix from a spiked population model[END_REF] (see also[START_REF] Onatski | Signal detection in high dimension: The multispiked case[END_REF]) considered a generalized problem which allowed for multiple spiked eigenvalues.

Note that x dB = 10 log 10 x. These are the typical units used for expressing SNR in wireless communication systems.

Note that under H 0 , μR = 0.

That is, when performing the test, the same decision threshold was chosen, as based on the asymptotic Gaussian distribution under H 0 .

-S 2 (z)[START_REF] Smith | Capacity of MIMO systems with semicorrelated flat fading[END_REF] withS 1 (z) = -λ 2 σ 2 F -2λμ F (z) -A F,1(z)

Acknowedgements

Thanks to Prof. Iain Johnstone at Stanford University for useful discussions in relation to the non-central multivariate F matrices (Model C) and for pointing out associated applications.

The work of D. Passemier and M. R. McKay was supported by the Hong Kong Research Grants Council (RGC) under grant number 616911. The work of Y. Chen was supported by the FDCT grant 077/2012/A3.

Finally, plugging [START_REF] Kang | Capacity of MIMO Rician channels[END_REF] and ( 64) into [START_REF] Taricco | Asymptotic mutual information statistics of separately correlated Rician fading MIMO channels[END_REF], we can rewrite [START_REF] Moustakas | Random matrix theory of multi-antenna communications: The Rician channel[END_REF] as

as n → ∞ such that m 1 /n → c 1 and m 2 /n → c 2 , where the function r(z 0 ) doesn't depend on λ. We recognise this as the MGF of the Gaussian distribution with mean nµ F + μF (z 0 ) and variance σ 2 F .

Proof of Corollary 1 and 2

Consider the expressions for μ(z 0 ) in [START_REF] Chen | On the linear statistics of Hermitian random matrices[END_REF] and μF (z 0 ) in [START_REF] Bai | Testing linear hypotheses in high-dimensional regressions[END_REF]. The term between brackets within the integrand is

For Model A, replacing z 0 by its value [START_REF] Lytova | Central limit theorem for linear eigenvalue statistics of random matrices with independent entries[END_REF] and the square root of the numerator by its value [START_REF] Anderson | An Introduction to Multivariate Statistical Analysis, 3rd Edition[END_REF], and setting δ = 0 , we obtain I = 0. Similarly using ( 8) and [START_REF] Telatar | Capacity of multi-antenna Gaussian channels[END_REF] for Model B, ( 14) and ( 68) for Model C, and setting ν = 0, we obtain I = 0. Thus, μ(z 0 ) = 0 and μF (z 0 ) = 0.

Appendix A. Statement and Proof of Lemma 2 Lemma 2. Let u > 0 and v > 0 such that nu + 1 > 0 and n(vu) / ∈ N. Assume that ℜz > 1 and γ ≥ 0. As n → ∞, we have

where

.

Proof. Under the prescribed conditions on u and v, we may use the following integral representation [87]

where the contour C starts at 0, traverses anti-clockwise around 1 and returns to 0. For large n, the Laplace approximation yields

for which

and where t 0 is the saddlepoint, which is the solution to

Thus, t 0 must satisfy

There is one solution which lies outside the contour for ℜz > 1:

Furthermore, we have

so that, taking the root with the correct phase factor (see [START_REF] Olver | Asymptotics and Special Functions[END_REF]Chapter 4] or [START_REF] Bleistein | Asymptotic Expansions of Integrals[END_REF]Chapter 7] for more details), we get 1

.

Substituting this quantity with (A.4) into (A.3) together with (A.2), we find the desired result (A.1). For the derivations of our results, we will require numerous integrals; these are summarized in (C.1)-(C.17). Note that for all definite integrals involving the variable t, these

Appendix B. Equivalence Between


Daniel Barlet
email: daniel.barlet@univ-lorraine.fr

Barlet Daniel

1 Introduction 2 2 The λ-connection. 2 2.1 The general situation. . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 The case of a polynomial with n + 2 monomials in n + 1 variables. 5 3 Two families of examples with d = 2 and h = 1. 9 3.1 The family x 2u + y 2v + z 2w + λ.x u .y v z w . . . . . . . . . . . . . . . . . 9 3.2 The family x 2p .z u + y 2q .z v + z u+v + λ.x p .y q . . . . . . . . . . . . . . .

Introduction

This note is a complement to the study in [B.13] of period integrals of non quasihomogeneous polynomials in n + 1 variables with n + 2 monomials. We focuse here on the dependance of these period integrals on the "natural" parameter λ ∈ C * which is the only "free" coefficient of such a polynomial modulo the dilatations of the variables1 For that purpose we recall first the fact that for a polynomial function f depending polynomially of a parameter λ we may define a natural "b-connection" on the highest (f, λ)-relative de Rham cohomology group of f which induces the derivation ∂ ∂λ on period integrals. The construction for any holomorphic function depending of a holomorphic parameter is precised in the appendix. Then we show how to compute explicitely this connection in our specific situation and we obtain a simple partial differential equation for the period integrals associated to any monomial in C[x 0 , . . . , x n ] when we consider a polynomial of the type

f λ (x) = n+1 j=1 x α j + λ.x α n+2 where α j ∈ N n+1 , j ∈ [1, n + 2]
with he following assumptions i) The (n + 2, n + 2)-matrix obtained from M := (α 1 , . . . , α n+2 ) by adding a first line of 1 has rank n + 2.

ii) The elements α 1 , . . . , α n+1 form a basis of Q n+1 .

Note that the first condition is equivalent to the fact that f is not quasi-homogeneous, and that the condition ii) is always satisfied assuming i), up to change the order of the monomials (and then to change the parameter λ to c.λ m for some c ∈ Q * some m ∈ Z * ).

2 The λ-connection.

2.1 The general situation.

We consider here a polynomial

f ∈ R := C[x 0 , . . . , x n ][λ]
f := Ω n+1 / d / f ∧ d / Ω n-1 / has a natural left A[λ]-module structure defined by
• The action of a is given by the multiplication by f .

• The action of b is given by d

/ f ∧ d -1 / .
Remark that for fixed λ, assuming that f λ has an isolated singularity at 0, the b-completion

E f ⊗ C[b] C[[b]]
Ã.[µ.dx] in E f ⊗ C[b] C[[b]
] and it determines a finite set of possible eigenvalues for the monodromies around s = 0 for the period integrals (λ fixed)

ϕ λ (s) = γ λ,s µ.dx d / f
for any horizontal family γ λ,s of compact n-dimensional cycles in the fibers of f λ .

It is important to remark that if f λ has a non isolated singularity at the origin, despite the fact that there is no finiteness for the C

[[b]]-module E f ⊗ C[b] C[[b]],
the conclusion above is still valid because the quotient à Ã.P (µ), and so its image in

E f ⊗ C[b] C[[b]], is a finite type C[[b]]-module.
Then the product decomposition

P d = (a -r 1 .b) . . . (a -r d .b
), where r 1 , . . . , r d are (explicitely computable) rational numbers, gives that the set {e 2iπ.r 1 , . . . , e 2iπ.r d } contains the spectrum of these monodromies (counting multiplicities).

Question. Is it true that P d is equal to the Bernstein element2 of the Brieskorn module Ã.

[µ.dx] in E f ⊗ C[b] C[[b]
] when f λ has an isolated singularity at the origin ?

Proposition 2.1.1 There exists a C-linear operator ∇ : E f → E f with the following properties :

1. For ω = d / ξ ∈ Ω n+1 / we have ∇([ω]) = [d / f ∧ ∂ξ ∂λ -∂f ∂λ .ω].
2. The map b -1 .∇ well defined on b. Ẽf where Ẽf := E f (b -torsion), with value in Ẽf , commutes with a and b and is a λ-connection.

If

(γ s,λ ) (s,λ)∈S×Ω is a horizontal family of compact n-cycles in the fibers of (f, λ) over an open set in C × C * \ C(f, λ) where C(f, λ) is the set of critical values of the map (f, id) : C n+1 × C * → C × C * , we have for any ω ∈ Ω n+1 / the equality ∂ ∂s ∂ ∂λ γ s,λ ω d / f = γ s,λ ∇(ω) d / f .
Proof. Remark first that ∇ is well defined because for ξ = d / η we have

∇(d / ξ) = d / f ∧ ∂(d / η) ∂λ = d / f ∧ d / ( ∂η ∂λ ) so it induces 0 in E f . Let ω = d / ξ ∈ Ω n+1 / and let d / η = d / f ∧ ξ. Then we have ∇(b.[ω]) = ∇(d / η) = d / f ∧ ∂η ∂λ - ∂f ∂λ .d / η = d / f ∧ ( ∂η ∂λ - ∂f ∂λ .ξ) = b d / ( ∂η ∂λ - ∂f ∂λ .ξ) = b ∂ d / f ∂λ ∧ ξ + d / f ∧ ∂ξ ∂λ - ∂f ∂λ .d / ξ -d / ( ∂f ∂λ ) ∧ ξ) = b d / ( ∂f ∂λ ).ξ + ∇(d / ξ) -d / ( ∂f ∂λ ) ∧ ξ) = b ∇(d / ξ)
as d / and ∂ ∂λ commute. So we have b.∇ = ∇.b. We have also

∇(a.[ω]) = ∇(f.d / ξ) = ∇(d / (f.ξ)) -∇(d / f ∧ ξ) = d / f ∧ ∂(f ξ) ∂λ - ∂f ∂λ .f.d / ξ - ∂f ∂λ .d / f ∧ ξ -∇(b.[ω]) = a.∇([ω]) -b.∇([ω]).
This implies the equality a.b -1 .∇ = b -1 .∇.a as C-linear maps from b. Ẽf to Ẽf .

Note that the equalities ∇.b = b.∇ and ∇.a = (a -b).∇ as C-endomorphisms of E f are more precise than the relations above.

Finally let ϕ ∈ C[λ] then we have

∇(ϕ.d / ξ) = ∇(d / (ϕ.ξ)) = d / f ∧ ∂ϕ.ξ ∂λ - ∂f ∂λ .ϕ.d / ξ = ∂ϕ ∂λ .(d / f ∧ ξ) + ϕ.∇(d / ξ) = ∂ϕ ∂λ .b[d / ξ] + ϕ.∇(d / ξ)
and this shows that b -1 .∇ is a λ-connection.

Note again that we proved the equality in E f : ∇(ϕ.ω) = ∂ϕ ∂λ .b.ω + ϕ.∇(ω) valid for ϕ ∈ C[λ] and ω ∈ E f which is more precise.

To prove the point 3. of the statement consider ξ ∈ Ω n / and let d be the total de Rham differential (in x and λ). We have

dξ = dλ ∧ ∂ξ ∂λ + d / ξ and df = dλ. ∂f ∂λ + d / f. Assume we can write dξ = dλ ∧ v + d / f ∧ u with u, v ∈ Ω n / .
Then we obtain

dξ = dλ ∧ (v - ∂f ∂λ .u) + d / f ∧ u with u = d / ξ d / f and v = ∂ξ ∂λ . If (γ s,λ
) is a horizontal family of compact n-cycles in the fibers of the map (f, id) :

C n+1 × C * → C × C * , we shall have d γ s,λ ξ = γ s,λ (v - ∂f ∂λ .u) .dλ + γ s,λ u .ds.
So, has the chain ∪ s,λ γ s,λ is proper and without λ-relative boundary we obtain

∂ ∂s γ s,λ ξ = γ s,λ d / ξ d / f and ∂ ∂λ γ s,λ ξ = γ s,λ ∂ξ ∂λ - ∂f ∂λ . d / ξ d / f Now consider ω ∈ Ω n+1 / and write ω = d / ξ. Then b[ω] = [d / f ∧ ξ] and we have γ s,λ b[ω] d / f = γ s,λ ξ and ∂ ∂s γ s,λ b[ω] d / f = γ s,λ ω d / f .
So we conclude that we have

∂ ∂λ γ s,λ ω d / f = ∂ ∂λ γ s,λ ξ = γ s,λ ∂ξ ∂λ - ∂f ∂λ . d / ξ d / f = ∂ ∂λ γ s,λ ∇[ω] d / f
2.2 The case of a polynomial with n + 2 monomials in n + 1 variables.

So we consider now the case were f := n+2 j=1 m j where m j := x α j j ∈ [1, n + 1] and m n+2 := λ.x α n+2 with the following hypotheses (see [B. 13]) : the rank of the square matrix M ′ := (α 1 , . . . , α n+1 ) is n + 1 and the rank of the square matrix M obtained by adding a first line of 1 to the matrix M := (α 1 , . . . , α n+2 ) is n + 2.

Recall that if we write (with a minimal positive integer r) r.α n+2 = n+1 j=1 p j .α j where p 1 , . . . , p n+1 are in Z, and if we define

d = inf{r - j,p j ≤0 p j , j,
E f is exactely the image in E f of C[m 1 , . . . , m n+2 ][λ].dx ⊂ Ω n+1 / with m j = x α j with j ∈ [1, n + 1] and m n+2 = λ.x α n+2 .
Our next result uses the following easy lemma:

Lemme 2.2.1 Let Q ∈ A a homogeneous element in (a, b) of degree k. Then for any λ ∈ C we have : b.Q.b -1 .(a -λ.b) = (a -(λ + k).b).Q.
proof. Remark first that the map A → A sending x ∈ A to b.x.b -1 is well defined and bijective thanks to the following facts : b is injective and b.A = A.b. We shall prove the lemma by induction k. As the case k = 0 is obvious, assume that the lemma is proved for k < k 0 where k 0 ≥ 1 and consider an homogeneous element Q of degree k 0 . We may assume4 that Q = b.R or that we may find µ ∈ C such that Q = (a -µ.b).R, where R is homogeneous of degree k 0 -1. In the first case we have, using the induction hypothesis :

b.b.R.b -1 .(a-λ.b) = b.(a-(λ+k 0 -1).b).R = (a-(λ+k 0 ).b).b.R = (a-(λ+k 0 ).b).Q.
In the second case we have, using the induction hypothesis :

b.(a -µ.b).R.b -1 .(a -λ.b) = (a -(µ + 1).b).b.R.b -1 .(a -λ.b) = (a -(µ + 1).b).(a -(λ + k 0 -1).R = (a -(λ + k 0 ).b).(a -µ.b).R = (a -(λ + k 0 ).b).Q. Proposition 2.2.2 Let µ be a monomial of degree k in C[x 0 , . . . , x n+1 ]. Then we have in E f the relation ∇([µ]) = -1 λ .(σ.a + (τ -k.σ).b)[µ]
where σ, τ are defined by the relation m n+2 .

[µ] = (σ.a + τ.b) [µ]. Moreover the value5 of σ is ±r/h so it does not depend on the choice of the monomial µ.

As a consequence, if we have on an open set S × Ω in C * × C * , a horizontal family (γ s,λ ) (s,λ)∈S×Ω of compact n-cycles in the fibers of the map

C n+1 × C * → C × C * defined by (x, λ) → (f λ (x), λ), the holomorphic function (s, λ) → ϕ(s, λ) := γ s,λ µ.dx d / f
satisfies the partial differential equation

-λ. ∂ ∂λ ∂ ∂s ϕ = σ. ∂(s.ϕ) ∂s + (τ -k).ϕ on S × Ω.
Proof. As we have λ.∇([1]) = -m n+2 in E f and as we know that there exist σ, τ in Q such that (σ.a + τ.b)[1] = m n+2 for the case µ = 1 the only thing to prove is the computation of σ. Using the Cramer system with matrix (n + 2, n + 2) obtained by adding a first line of 1 to the matrix M := (α 1 , . . . , α n+2 ), computing a[1] and the b i [1] we find that σ is the coefficient (n + 2, 1) in the matrix M-1 . Let M ′ be the principal (n + 1, n + 1) minor of M . This implies that σ = (-1) n+1 det(M ′ ) det( M) .

But using the relation α n+2 = n+1 j=1 p j r .α j we obtain

det( M ) = (-1) n+1 .(1 - n+1 j=1 p j r ).det(M ′ ) so we conclude that σ = r r -n+1 j=1 p j . Now we have two cases : i) r -p j <0 p j = d + h > p j >0 p j = d. Then r -n+1 j=1 p j = (d + h) -d = h. So σ = r/h, and the exponent of λ in P is r. ii) p j >0 p j = d + h > r -p j <0 p j = d. Then r -n+1 j=1 p j = d -(d + h) = -h. So σ = -r/h,
and the exponent of λ in P is -r.

Consider now the case of a degree

k monomial µ ∈ C[x 0 , . . . , x n ]. Then there exists again σ ′ , τ ′ in Q such that (σ ′ .a + τ ′ .b)[µ] = [m n+2 .µ] in E f . As a[µ], (β i + 1).b[µ], i ∈ [0, n]
, where β i is the degree in x i of µ := x β , are again given from the [m j .µ], j ∈ [1, n + 2] by the same Cramer system, we conclude that σ ′ = σ. To conclude the proof it is enough to apply the proposition 2.1.1. Note that in the case i) above P := P d+h + c.λ r .P d annihilated [µ] in E f and in the case ii) we have P := P d+h + c.λ -r .P d .

The lemma 2.2.1 gives that λ.∇(P. Remark. Recall that in [B.13] we have built in an explicit way a differential equation in s ∈ S, depending in a very simple and concrete way on λ ∈ C * which is satisfied by ϕ. So it is easy to see that the knowledge of a formal asymptotic expansion when s goes to 0 in S6 for a given λ 0 , of the type

ϕ(λ 0 , s) ≃ i,j C i,j .s ρ i .(Logs) j
where ρ 1 , . . . , ρ I are in -1 + Q * + , j ∈ [0, n] are integers and C i,j are in C[[s]], determines (uniquely) via the partial differential equation above, a formal expansion of the same type for each given λ ∈ Ω, whose coefficients C λ i,j are polynomials in Logλ easily computable from the coefficients C λ 0 i,j := C i,j of the asymptotic expansion at the initial value λ 0 of λ. This is described in the following lemma.

Lemme 2.2.3 Let Ω be a simply connected domain in C * . Let (ρ i ) i∈I be a finite collection of rational numbers strictly bigger than -1. Assume that the formal power serie 2. The generalization of this proposition to several holomorphic parameters is immediate.

ϕ λ := N k=0 i∈I m≥0 c i,k m (λ).
Bibliography.

•

  depending polynomially on a parameter λ. We consider on R ⊗ Λ * (C n+1 ) := Ω * / the λ-relative de Rham complex, where (C n+1 ) * := ⊕ n i=0 C.dx i , and we denote d / its differential. We shall denote by A the unitary (non commutative) algebra generated by a and b with the commutation relation a.b -b.a = b 2 and by A[λ] := A ⊗ C C[λ] with its natural structure of algebra for which λ commutes with a and b. Then the quotient E

  [µ]) = -(σ.a + (τ ′ -k.σ).b).P [µ] which makes explicit the fact that λ.∇ is well defined on A[λ].[µ] ⊂ E f .

  s m+ρ i .(Logs) k k! where c i,k m are holomorphic functions in Ω, satisfies the partial differential equation for each λ ∈ Ω. Then for each i, k fixed, the function c i,k m is a polynomial in Logλ of degree ≤ m for each m. Moreover the collection of numbers c i,k m (λ 0 ) for a given λ 0 ∈ Ω determines uniquely these polynomials. 2. The map b -1 .∇ well defined on b. H• where H• := H • (b -torsion), with value in H• , commutes with a and b and is a λ-connection. 3. If (γ s,λ ) (s,λ)∈S×Ω is a horizontal family of compact p-cycles in the fibers of (f, λ) over an open set in D × M \ C(f, λ) where C(f, λ) is the set of critical values of the map (id, f ) : D × M → D × C, we have for any ω First remark that if ω= d / ξ ∈ K p+1 with d / f ∧ ξ = 0, we have d / ( ∂f ∂λ ) ∧ ξ + d / f ∧ ∂ξ ∂λ = 0 so ∇(d / ξ) = -d / ( ∂f ∂λ .ξ) is in d / K p . Now for ω = d / ξ ∈ K p+1 we have d / f ∧ ∇(d / ξ) = 0 and d / ∇(d / ξ) = -d / f ∧ d / ∂ξ ∂λ -d / ∂f ∂λ ∧ d / ξand we obtain thatd / ∇(d / ξ) = -∂ ∂λ d / f ∧ ω = 0 using the fact that ∂ ∂λ d / f ∧ ω ≡ 0.The proof of the other statements are analogous to the corresponding ones in proposition 2.1.1.Remarks.1. As above we have a more precise formulation for the properties in assertion 2. of the proposition above with the following relations in H • ∇(ϕ.ω) = ∂ϕ ∂λ .b.ω + ϕ.∇(ω) for ϕ ∈ O λ and ω ∈ H • b.∇ = ∇.b and ∇.a = (a -b).∇.

  [B.08] Barlet,D. Sur certaines singularités d'hypersurfaces II, Journal of Algebraic Geometry 17 (2008), p.199-254. • [B.09] Barlet,D. Périodes évanescentes et (a,b)-modules monogènes, Bollettino U.M.I. (9) II (2009) p.651-697. • [B.13] Barlet, D. Algebraic differential equations associated to some polynomials arXiv:1305.6778 ( math.arxiv 2013).

  P d+h + c.λ ±r .P d which annihilated the class [dx] in E f , where P d+h and P d are homogeneous elements in A, respectively of degree d + h and d which are monic in a with rational coefficients 3 , and where c is in Q * . The sign in the exponent of λ will be precised in the proof of the proposition 2.2.2. Recall also that in this situation the A[λ] module generated by the class [dx] in

	p j } and	d + h = sup{r -	p j ,	p j }
	p j >0	j,p j ≤0	j,p j >0
	there exists an element P in A[λ, λ -1 ] of the form
	P :=

In our hypothesis we may assume that all monomials have coefficient 1 excepted the last one up to a linear diagonal change of variable.

For a (a,b)-module E with one generator as a Ã-module, the relation between its Bernstein element P d ∈ A and its Bernstein polynomial B is given by the formula (see [B.09]) (-b) d .B(-b -1 .a) = P d where d is the rank of E.

More is proved in [B.13] : P d+h and P d factorize in product of (ar j .b) with r j ∈ Q.

Recall that any homogeneous element in A which is monic in a factorizes as a product of linear factors (ar i .b), where the r i are complex numbers; see[B.09].

The sign is precised in the proof and only depends on α 1 , . . . , α n+2 .

This is always the case when S contains an open sector with edge at the origin.

Proof. The partial differential equation implies the following recursion relation for each i, k, m : (m + ρ i + 1).λ. ∂c i,k m+1 (λ) ∂λ + λ. ∂c i,k+1 m+1 (λ) ∂λ = α.(m

We shall make a descending induction on k. For k = N the recursion relation reduces to

and an easy induction on m ≥ 0 gives our assertion for k = N. Assuming the statement proved for k + 1 a simple quadrature in λ implies the case k.

3 Two families of examples with d = 2 and h = 1.

The family x

The condition to be in our situation is u.v.w > 0. Then we have the relation m 2 4 = λ 2 .m 1 .m 2 .m 3 and it shows that d = 2 and h = 1. Note that the only singularity of f in {f = 0} is the origine. To compute P := P 3 + c.λ -2 .P 2 which annihilates [1] is not difficult. We find

In this case we have

Here we are in the case ii) above (so σ = -2).

Let me illustrate this family on a simple example :

In this case we find

3.2 The family x 2p .z u + y 2q .z v + z u+v + λ.x p .y q .

The condition to be in our situation is p.q.(u + v) > 0. Note that the singularity at the origine is not isolated in general in these cases. We have here the equality

The relation which determines P annihilating [1] is given by m 2 4 .m 3 = λ 2 .m 1 .m 2 , so r = 2, d = 2, h = 1 and we are in the case i). The computation of P gives

The computation of (σ, τ ) such that (σ.a + τ.b)[1] = m 4 is easy and it gives

Again let me illustrated by an example : for p = q = 2 and u = v = 1 so for f = x 4 .z + y 4 .z + z 2 + λ.x 2 .y 2 . We find

4 Appendix

It is interesting to remark that the proposition 2.1.1 is a special case in a specific algebraic setting of a general result on the the filtered Gauss-Manin connexion of a holomorphic function depending holomorphically of a parameter. This is the goal of this appendix to precise this point.

Let Denote by (K • / , d / ) the topological restriction of the λ-relative de Rham complex (defined above) for the map (λ, x) → (λ, f (λ, x)), to the analytic subset

and let H p the p-th cohomology sheaf of this complex. Recall that these cohomogy sheaves have a natural structure of left A[λ]-modules with the action of a given by the multiplication by f and with the action of b defined by d / f ∧ d -1 / .

Proposition 4.0.1 There exists a natural graded map ∇ • : H • → H • with the following properties :