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

الأصل · EN

We study the joint limit distribution of the k largest eigenvalues of a p× p sample covariance matrix XX based on a large p× n matrix X. The rows of X are given by independent copies of a linear process, Xit=∑ⱼ cⱼ Zᵢ,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 XX 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).