Synchronous code-division multiple-access communication systems with randomly chosen spreading sequences and capacity-achieving forward error correction coding are analyzed in terms of spectral efficiency. Emphasis is on the penalties paid by applying single user coding in conjuction with suboptimal multiuser receivers as opposed to optimal joint decoding which involves complexity that is exponential in the number of users times the codeword length. The conventional, the decorrelating and the (re-encoded) decorrelating decision-feedback detectors are analyzed in the nonasymptotic case for spherical random sequences. The re-encoded minimum mean squared error (MMSE) decision-feedback receiver achieving the same performance as joint multiuser decoding for equal power users is shown to be suboptimal in the case of equal rates.

Abstract. Consider a N × n matrix Zn = (Zn) where the individual entries are a j1j2 realization of a properly rescaled stationary gaussian random field: Z n 1 ∑ j1j2 = √ h(k1, k2)U(j1 − k1, j2 − k2), n (k1,k2)∈Z 2 where h ∈ ℓ1 (Z2) is a deterministic complex summable sequence and (U(j1, j2);(j1, j2) ∈ Z2) is a sequence of independent complex gaussian random variables with mean zero and unit variance. The purpose of this article is to study the limiting empirical distribution of the eigenvalues of Gram random matrices such as ZnZ ∗ n and (Zn + An)(Zn + An) ∗ where An is a deterministic matrix with appropriate assumptions in the case where n → ∞ and N n → c ∈ (0, ∞). The proof relies on related results for matrices with independent but not identically distributed entries and substantially differs from related works in the literature (Boutet de Monvel et al. [3], Girko [7], etc.).

We place ourselves in the setting of high-dimensional statistical inference, where the number of variables p in a dataset of interest is of the same order of magnitude as the number of observations n. More formally we study the asymptotic properties of correlation and covariance matrices under the setting that p/n → ρ ∈ (0, ∞), for general population covariance. We show that spectral properties for large dimensional correlation matrices are similar to those of large dimensional covariance matrices, for a large class of models studied in random matrix theory. We also derive a Marčenko-Pastur type system of equations for the limiting spectral distribution of covariance matrices computed from data with elliptical distributions and generalizations of this family. The motivation for this study comes partly from the possible relevance of such distributional assumptions to problems in econometrics and portfolio optimization, as well as robustness questions for certain classical random matrix results. A mathematical theme of the paper is the important use we make of concentration inequalities.

Consider an N × n random matrix Yn = (Y n ij) where the entries are given by Y n ij = σij(n) √ X n 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 ×n matrix An whose columns and rows are uniformly bounded in the Euclidean norm. Let Σn = Yn + An. We prove in this article that there exists a deterministic N ×N matrix-valued function Tn(z) analytic in C −R + such that, almost surely, 1 lim

Models where the number of parameters increases with the sample size, are becoming increasingly important in statistics. This necessitates a close look at the statistical properties of eigenvalues of random matrices whose dimension increases indefinitely. There are several properties of the eigenvalues that one would be interested in and the literature in this area is already huge. In this article we focus on one important aspect: the existence and identification of the limiting spectral distribution (LSD) of the empirical distribution of the eigenvalues. We describe some of the general tools used in establishing the LSD and how they have been applied successfully to establish results on the LSD for certain types of matrices. Some of the matrices for which the LSD has been established and the nature of the limit laws known are described in detail. We also discuss a few open problems and partial solutions for some of these. We introduce a few new ideas which seem to hold some promise in this area. We also establish an invariance result for random Toeplitz matrix.

Abstract. We consider random hermitian matrices in which distant abovediagonal entries are independent but nearby entries may be correlated. We find the limit of the empirical distribution of eigenvalues by combinatorial methods. We also prove that the limit has algebraic Stieltjes transform by an argument based on dimension theory of noetherian local rings. 1.

Abstract. Consider a N × n random matrix Yn = (Y n ij) where the entries are given by Y n σij(n) ij = √ X n n ij, the Xn ij being centered, independent and identically distributed random variables with unit variance and (σij(n); 1 ≤ i ≤ N,1 ≤ j ≤ 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 (YnY ∗ n + ρIN) where Y ∗ is the Hermitian adjoint of Y and ρ> 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 th moment of the Xij’s differs from the 4 th moment of a Gaussian random variable. Such a CLT is of interest in the field of wireless communications. Key words and phrases: Random Matrix, empirical distribution of the eigenvalues, Stieltjes

a separable covariance matrix

We study classical ensembles of real symmetric random matrices introduced by Eugene Wigner. We discuss Stein’s method for the asymptotic approximation of expectations of functions of the normalized eigenvalue counting measure of high dimensional matrices. The method is based on a differential equation for the density of the semi-circular law.

Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of An and Bn is obtained and studied.