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

We study the joint limit distribution of the k largest eigenvalues of a p × p sample covariance matrix X X T based on a large p × n matrix X . The rows of X are given by independent copies of a linear process, X i t = ∑ j c j Z i , t − j , with regularly varying noise ( Z i t ) with tail index α ∈ ( 0 , 4 ) . It is shown that a point process based on the eigenvalues of X X 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 j 2 . This result is extended to random coefficient models where the coefficients of the linear processes ( X i t ) 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 ) .