Eigenvalue variance bounds for covariance matrices

This work is concerned with finite range bounds on the variance of individual eigenvalues of random covariance matrices, both in the bulk and at the edge of the spectrum. In a preceding paper, the author established analogous results for Wigner matrices [7] and stated the results for covariance matrices. They are proved in the present paper. Relying on the LUE example, which needs to be investigated first, the main bounds are extended to complex covariance matrices by means of the Tao, Vu and Wang Four Moment Theorem and recent localization results by Pillai and Yin. The case of real covariance matrices is obtained from interlacing formulas. Random covariance matrices, or Wishart matrices, were introduced by the statistician Wishart in 1928 to model tables of random data in multivariate statistics. The spectral properties of these matrices are indeed useful for example for studying the properties of certain random vectors, elaborating statistical tests and for principal component analysis. Similarly to Wigner matrices, which were introduced by the physicist Wigner in the fifties in order to study infinite-dimensional operators in statistical physics, the asymptotic spectral properties were soon conjectured to be universal in the sense they do not depend on the distribution of the entries (see for example [1] and [19]). Eigenvalues were studied asymptotically both at the global and local regimes, considering for instance the global behavior of the spectrum, the behavior of extreme eigenvalues or the spacings between eigenvalues in the bulk of the spectrum. In the Gaussian case, the eigenvalue joint distribution is explicitly known, allowing for a complete study of the asymptotic spectral properties (see for example [1], [3], [21]). One of the main goals of random matrix theory over the past decades was to extend these results to non-Gaussian covariance matrices. 1 ha l-0 08 65 62 8, v er si on 1 24 S ep 2 01 3 However, in multivariate statistics, quantitative finite-range results are more useful than asymptotic properties. Furthermore, random covariance matrices have become useful in several other fields, such as compressed sensing (see [31]), wireless communication and quantitative finance (see [3]). In these fields too, quantitative results are of high interest. Several recent developments have thus been concerned with non-asymptotic random matrix theory. See for example some recent surveys and papers on this topic [24], [31] and [30]. In this paper, we investigate in this respect variance bounds on the eigenvalues of families of covariance matrices. In a preceding paper [7], we established similar bounds for Wigner matrices and the results for covariance matrices were stated but not proved. In the present paper, we provide the corresponding proofs. For the sake of completeness and in order to make the present paper readable separately, we reproduce here some parts of the previous one [7]. Random covariance matrices are defined by the following. Let X be a m × n (real or complex) matrix, with m > n, such that its entries are independent, centered and have variance 1. Then Sm,n = 1 m X∗X is a covariance matrix. An important example is the case when the entries of X are Gaussian. Then Sm,n belongs to the so-called Laguerre Unitary Ensemble (LUE) if the entries of X are complex and to the Laguerre Orthogonal Ensemble (LOE) if they are real. Sm,n is Hermitian (or real symmetric) and therefore has n real eigenvalues. As m > n, none of these eigenvalues is trivial. Furthermore, these eigenvalues are nonnegative and will be denoted by 0 6 λ1 6 · · · 6 λn. Among universality results, the classical Marchenko-Pastur theorem states that, if m n → ρ > 1 when n goes to infinity, the empirical spectral measure Lm,n = 1 n ∑n j=1 δλj converges almost surely to a deterministic measure, called the Marchenko-Pastur distribution of parameter ρ. This measure is compactly supported and is absolutely continuous with respect to Lebesgue measure, with density dμMP (ρ)(x) = 1 2πx √ (bρ − x)(x− aρ)1[aρ ,bρ](x)dx, where aρ = (1−√ρ)2 and bρ = (1+√ρ)2 (see for example [3]). We denote by μm,n the approximate Marchenko-Pastur density μm,n(x) = 1 2πx √ (x− am,n)(bm,n − x)1[am,n ,bm,n](x), with am,n = (