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

作者：Davis Richard A; Pfaffel Oliver^*; Stelzer Robert

来源：Stochastic Processes and Their Applications, 2014, 124(1): 18-50.

DOI：10.1016/j.spa.2013.07.005

• Random matrix theory
• Heavy-tailed distribution
• Random matrix with dependent entries
• Largest singular value
• Sample covariance matrix
• Largest eigenvalue
• Linear process
• Random coefficient model

We study the joint limit distribution of the k largest eigenvalues of a p x p sample covariance matrix XX inverted perpendicular based on a large p x n matrix X. The rows of X are given by independent copies of a linear process, X-it = Sigma(j) C-i Z(i,j-i), with regularly varying noise (Z(it)) with tail index a alpha is an element of (0, 4). It is shown that a point process based on the eigenvalues of XX inverted perpendicular converges, as n -> infinity and p -> infinity at a suitable rate, in distribution to a Poisson point process with an intensity measure depending on alpha and Sigma c(j)(2). This result is extended to random coefficient models where the coefficients of the linear processes (X-it) are given by c(j) (theta(i)) for some ergodic sequence (theta(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 alpha is an element of (0, 2).

• 出版日期2014-1