On the Rate of Convergence to the Marchenko–Pastur Distribution
نویسندگان
چکیده
LetX = (Xjk) denote n×p random matrix with entriesXjk, which are independent for 1 ≤ j ≤ n, 1 ≤ k ≤ p. We consider the rate of convergence of empirical spectral distribution function of matrix W = 1 p XX ∗ to the Marchenko–Pastur law. We assume that EXjk = 0, EX 2 jk = 1 and that the distributions of the matrix elements Xjk have a uniformly sub exponential decay in the sense that there exists a constant κ > 0 such that for any 1 ≤ j ≤ n, 1 ≤ k ≤ p and any t ≥ 1 we have Pr{|Xjk| > t} ≤ κ−1 exp{−t}. By means of a recursion argument it is shown that the Kolmogorov distance between the empirical spectral distribution of the sample covariance matrix W and the Marchenko–Pastur distribution is of order O(n−1 log n) with some positive constant b > 0.
منابع مشابه
The rate of convergence of spectra of sample covariance matrices
It is shown that the Kolmogorov distance between the spectral distribution function of a random covariance matrix 1 p XX T , where X is a n × p matrix with independent entries and the distribution function of the Marchenko-Pastur law is of order O(n −1/2). The bounds hold uniformly for any p, including p n equal or close to 1.
متن کاملOn a surprising relation between the Marchenko-Pastur law, rectangular and square free convolutions
In this paper, we prove a result linking the square and the rectangular R-transforms, which consequence is a surprising relation between the square and rectangular free convolutions, involving the Marchenko-Pastur law. Consequences on infinite divisibility and on the arithmetics of Voiculescu’s free additive and multiplicative convolutions are given.
متن کاملOn a question of Babadi and Tarokh II
In this paper we continue to study a question proposed by Babadi and Tarokh [4] on the mysterious randomness of Gold sequences. Upon improving their result, we establish the randomness of product of pseudorandom matrices formed from two linear block codes with respect to the empirical spectral distribution, if the dual distance of both codes is at least 5, hence providing an affirmative answer ...
متن کاملMarchenko-Pastur Law for Tyler’s and Maronna’s M-estimators
This paper studies the limiting behavior of Tyler’s and Maronna’s Mestimators, in the regime that the number of samples n and the dimension p both go to infinity, and p/n converges to a constant y with 0 < y < 1. We prove that when the data samples are identically and independently generated from the Gaussian distribution N(0, I), the difference between the sample covariance matrix and a scaled...
متن کامل