Gaussian fluctuations for linear spectral statistics of large random covariance matrices

Consider a $N\times n$ matrix $\Sigma_n=\frac{1}{\sqrt{n}}R_n^{1/2}X_n$, where $R_n$ is a nonnegative definite Hermitian matrix and $X_n$ is a random matrix with i.i.d. real or complex standardized entries. The fluctuations of the linear statistics of the eigenvalues $$\operatorname {Trace}f \bigl(\Sigma_n\Sigma_n^*\bigr)=\sum_{i=1}^Nf(\lambda_i),\qquad (\lambda_i)\ eigenvalues\ of\ \Sigma_n\Sigma_n^*,$$ are shown to be Gaussian, in the regime where both dimensions of matrix $\Sigma_n$ go to infinity at the same pace and in the case where $f$ is of class $C^3$, that is, has three continuous derivatives. The main improvements with respect to Bai and Silverstein's CLT [Ann. Probab. 32 (2004) 553-605] are twofold: First, we consider general entries with finite fourth moment, but whose fourth cumulant is nonnull, that is, whose fourth moment may differ from the moment of a (real or complex) Gaussian random variable. As a consequence, extra terms proportional to $\vert \mathcal{V}\vert ^2=\bigl|\mathbb{E}\bigl(X_{11}^n\bigr) ^2\bigr|^2$ and $\kappa=\mathbb{E}\bigl \vert X_{11}^n\bigr \vert ^4-\vert {\mathcal{V}}\vert ^2-2$ appear in the limiting variance and in the limiting bias, which not only depend on the spectrum of matrix $R_n$ but also on its eigenvectors. Second, we relax the analyticity assumption over $f$ by representing the linear statistics with the help of Helffer-Sj\"{o}strand's formula. The CLT is expressed in terms of vanishing L\'{e}vy-Prohorov distance between the linear statistics' distribution and a Gaussian probability distribution, the mean and the variance of which depend upon $N$ and $n$ and may not converge.Comment: Published at http://dx.doi.org/10.1214/15-AAP1135 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Deterministic equivalents for certain functionals of large random matrices

Consider an $N\times n$ random matrix $Y_n=(Y^n_{ij})$ where the entries are given by $Y^n_{ij}=\frac{\sigma_{ij}(n)}{\sqrt{n}}X^n_{ij}$, the $X^n_{ij}$ being independent and identically distributed, centered with unit variance and satisfying some mild moment assumption. Consider now a deterministic $N\times n$ matrix A_n whose columns and rows are uniformly bounded in the Euclidean norm. Let $\Sigma_n=Y_n+A_n$. We prove in this article that there exists a deterministic $N\times N$ matrix-valued function T_n(z) analytic in $\mathbb{C}-\mathbb{R}^+$ such that, almost surely, $$\lim_{n\to+\infty,N/n\to c}\biggl(\frac{1}{N}\operatorname {Trace}(\Sigma_n\Sigma_n^T-zI_N)^{-1}-\frac{1}{N}\operatorname {Trace}T_n(z)\biggr)=0.$$ Otherwise stated, there exists a deterministic equivalent to the empirical Stieltjes transform of the distribution of the eigenvalues of $\Sigma_n\Sigma_n^T$. For each n, the entries of matrix T_n(z) are defined as the unique solutions of a certain system of nonlinear functional equations. It is also proved that $\frac{1}{N}\operatorname {Trace} T_n(z)$ is the Stieltjes transform of a probability measure $\pi_n(d\lambda)$, and that for every bounded continuous function f, the following convergence holds almost surely $$\frac{1}{N}\sum_{k=1}^Nf(\lambda_k)-\int_0^{\infty}f(\lambda)\pi _n(d\lambda)\mathop {\longrightarrow}_{n\to\infty}0,$$ where the $(\lambda_k)_{1\le k\le N}$ are the eigenvalues of $\Sigma_n\Sigma_n^T$. This work is motivated by the context of performance evaluation of multiple inputs/multiple output (MIMO) wireless digital communication channels. As an application, we derive a deterministic equivalent to the mutual information: $$C_n(\sigma^2)=\frac{1}{N}\mathbb{E}\log \det\biggl(I_N+\frac{\Sigma_n\Sigma_n^T}{\sigma^2}\biggr),$$ where $\sigma^2$ is a known parameter.Comment: Published at http://dx.doi.org/10.1214/105051606000000925 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

A CLT for Information-theoretic statistics of Gram random matrices with a given variance profile

Consider a $N\times n$ random matrix $Y_n=(Y_{ij}^{n})$ where the entries are given by $$Y_{ij}^{n}=\frac{\sigma_{ij}(n)}{\sqrt{n}} X_{ij}^{n}$$ the $X_{ij}^{n}$ being centered, independent and identically distributed random variables with unit variance and $(\sigma_{ij}(n); 1\le i\le N, 1\le j\le n)$ being an array of numbers we shall refer to as a variance profile. We study in this article the fluctuations of the random variable $$\log\det(Y_n Y_n^* + \rho I_N)$$ where $Y^*$ is the Hermitian adjoint of $Y$ and $\rho > 0$ is an additional parameter. We prove that when centered and properly rescaled, this random variable satisfies a Central Limit Theorem (CLT) and has a Gaussian limit whose parameters are identified. A complete description of the scaling parameter is given; in particular it is shown that an additional term appears in this parameter in the case where the 4$^\textrm{th}$ moment of the $X_{ij}$'s differs from the 4$^{\textrm{th}}$ moment of a Gaussian random variable. Such a CLT is of interest in the field of wireless communications

Large Complex Correlated Wishart Matrices: The Pearcey Kernel and Expansion at the Hard Edge

We study the eigenvalue behaviour of large complex correlated Wishart matrices near an interior point of the limiting spectrum where the density vanishes (cusp point), and refine the existing results at the hard edge as well. More precisely, under mild assumptions for the population covariance matrix, we show that the limiting density vanishes at generic cusp points like a cube root, and that the local eigenvalue behaviour is described by means of the Pearcey kernel if an extra decay assumption is satisfied. As for the hard edge, we show that the density blows up like an inverse square root at the origin. Moreover, we provide an explicit formula for the $1/N$ correction term for the fluctuation of the smallest random eigenvalue.Comment: 40 pages, 6 figures. Accepted for publication in EJ

Estimation of the Covariance Matrix of Large Dimensional Data

This paper deals with the problem of estimating the covariance matrix of a series of independent multivariate observations, in the case where the dimension of each observation is of the same order as the number of observations. Although such a regime is of interest for many current statistical signal processing and wireless communication issues, traditional methods fail to produce consistent estimators and only recently results relying on large random matrix theory have been unveiled. In this paper, we develop the parametric framework proposed by Mestre, and consider a model where the covariance matrix to be estimated has a (known) finite number of eigenvalues, each of it with an unknown multiplicity. The main contributions of this work are essentially threefold with respect to existing results, and in particular to Mestre's work: To relax the (restrictive) separability assumption, to provide joint consistent estimates for the eigenvalues and their multiplicities, and to study the variance error by means of a Central Limit theorem

A Central Limit Theorem for the SINR at the LMMSE Estimator Output for Large Dimensional Signals

This paper is devoted to the performance study of the Linear Minimum Mean Squared Error estimator for multidimensional signals in the large dimension regime. Such an estimator is frequently encountered in wireless communications and in array processing, and the Signal to Interference and Noise Ratio (SINR) at its output is a popular performance index. The SINR can be modeled as a random quadratic form which can be studied with the help of large random matrix theory, if one assumes that the dimension of the received and transmitted signals go to infinity at the same pace. This paper considers the asymptotic behavior of the SINR for a wide class of multidimensional signal models that includes general multi-antenna as well as spread spectrum transmission models. The expression of the deterministic approximation of the SINR in the large dimension regime is recalled and the SINR fluctuations around this deterministic approximation are studied. These fluctuations are shown to converge in distribution to the Gaussian law in the large dimension regime, and their variance is shown to decrease as the inverse of the signal dimension

Performance of Statistical Tests for Single Source Detection using Random Matrix Theory

This paper introduces a unified framework for the detection of a source with a sensor array in the context where the noise variance and the channel between the source and the sensors are unknown at the receiver. The Generalized Maximum Likelihood Test is studied and yields the analysis of the ratio between the maximum eigenvalue of the sampled covariance matrix and its normalized trace. Using recent results of random matrix theory, a practical way to evaluate the threshold and the $p$-value of the test is provided in the asymptotic regime where the number $K$ of sensors and the number $N$ of observations per sensor are large but have the same order of magnitude. The theoretical performance of the test is then analyzed in terms of Receiver Operating Characteristic (ROC) curve. It is in particular proved that both Type I and Type II error probabilities converge to zero exponentially as the dimensions increase at the same rate, and closed-form expressions are provided for the error exponents. These theoretical results rely on a precise description of the large deviations of the largest eigenvalue of spiked random matrix models, and establish that the presented test asymptotically outperforms the popular test based on the condition number of the sampled covariance matrix.Comment: 45 p. improved presentation; more proofs provide

Fluctuations of an improved population eigenvalue estimator in sample covariance matrix models

This article provides a central limit theorem for a consistent estimator of population eigenvalues with large multiplicities based on sample covariance matrices. The focus is on limited sample size situations, whereby the number of available observations is known and comparable in magnitude to the observation dimension. An exact expression as well as an empirical, asymptotically accurate, approximation of the limiting variance is derived. Simulations are performed that corroborate the theoretical claims. A specific application to wireless sensor networks is developed.Comment: 30 p