Big n, Big p: Eigenvalues for Cov Matrices of Heavy-Tailed Multivariate Time Series

Abstract:

n this paper we give an asymptotic theory for the eigenvalues of the sample covariance matrix of a multivariate time series when the number of components p goes to infinity with the sample size. The time series constitutes a linear process across time and between components. The input noise of the linear process has regularly varying tails with index between 0 and 4; in particular, the time series has infinite fourth moment. We derive the limiting behavior for the largest eigenvalues of the sample covariance matrix and show point process convergence of the normalized eigenvalues as n and p go to infinity. The limiting process has an explicit form involving points of a Poisson process and eigenvalues of a non-negative definite matrix. Based on this convergence we derive limit theory for a host of other continuous functionals of the eigenvalues, including the joint convergence of the largest eigenvalues, the joint convergence of the largest eigenvalue and the trace of the sample covariance matrix, and the ratio of the largest eigenvalue to their sum.