Limit theory for the largest eigenvalues of sample covariance matrices with heavy-tails

Richard A. Davis, Oliver Pfaffel and Robert Stelzer

Stochastic Processes and their Applications, 2014, vol. 124, issue 1, 18-50

Abstract: We study the joint limit distribution of the k largest eigenvalues of a p×p sample covariance matrix XXT based on a large p×n matrix X. The rows of X are given by independent copies of a linear process, Xit=∑jcjZi,t−j, with regularly varying noise (Zit) with tail index α∈(0,4). It is shown that a point process based on the eigenvalues of XXT converges, as n→∞ and p→∞ at a suitable rate, in distribution to a Poisson point process with an intensity measure depending on α and ∑cj2. This result is extended to random coefficient models where the coefficients of the linear processes (Xit) are given by cj(θi), for some ergodic sequence (θi), and thus vary in each row of X. As a by-product of our techniques we obtain a proof of the corresponding result for matrices with iid entries in cases where p/n goes to zero or infinity and α∈(0,2).

Keywords: Random matrix theory; Heavy-tailed distribution; Random matrix with dependent entries; Largest singular value; Sample covariance matrix; Largest eigenvalue; Linear process; Random coefficient model (search for similar items in EconPapers)
Date: 2014
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (6)

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DOI: 10.1016/j.spa.2013.07.005

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