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

2011/08/27 by Davis, Richard A., Pfaffel, Oliver, Stelzer, Robert
#60B20 #62G32 #FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1108.5464

Abstract

We study the joint limit distribution of the k largest eigenvalues of a p× p sample covariance matrix XX^\T based on a large p× n matrix X. The rows of X are given by independent copies of a linear process, X_it=∑_j c_j Z_i,t-j, with regularly varying noise (Z_it) with tail index α∈(0,4). It is shown that a point process based on the eigenvalues of XX^\T converges, as n→∞ and p→∞ at a suitable rate, in distribution to a Poisson point process with an intensity measure depending on α and ∑ c_j². This result is extended to random coefficient models where the coefficients of the linear processes (X_it) are given by c_j(θ_i), for some ergodic sequence (θ_i), 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).

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