On the empirical spectral distribution for matrices with long memory and independent rows

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:126

Article

Merlevede, F.¹  Peligrad, M.² 

[1] Univ Paris Est, CNRS, UMR 8050, LAMA,UPEM,UPEC, Champs Sur Marne, France

[2] Univ Cincinnati, Dept Math Sci, POB 210025, Cincinnati, OH 45221 USA

关键词: Random matrices;    Stieltjes transform;    Martingale approximation;    Lindeberg method;    Empirical eigenvalue distribution;    Spectral density;    Sample covariance matrix;   

DOI  :  10.1016/j.spa.2016.02.016

来源: Elsevier

【 摘 要 】

In this paper we show that the empirical eigenvalue distribution of any sample covariance matrix generated by independent samples of a stationary regular sequence has a limiting distribution depending only on the spectral density of the sequence. We characterize this limit in terms of Stieltjes transform via a certain simple equation. No rate of convergence to zero of the covariances is imposed, so, the underlying process can exhibit long memory. If the stationary sequence has trivial left sigma field the result holds without any other additional assumptions. This is always true if the entries are functions of i.i.d. As a method of proof, we study the empirical eigenvalue distribution for a symmetric matrix with independent rows below the diagonal; the entries satisfy a Lindeberg-type condition along with mixingale-type conditions without rates. In this nonstationary setting we point out a property of universality, meaning that, for large matrix size, the empirical eigenvalue distribution depends only on the covariance structure of the sequence and is independent on the distribution leading to it. These results have interest in themselves, allowing to study symmetric random matrices generated by random processes with both short and long memory. (C) 2016 Elsevier B.V. All rights reserved.

【 授权许可】

Free