Eigenvalues and eigenvectors of heavy-tailed sample covariance matrices with general growth rates: the iid case

TL;DR

This paper extends the understanding of the asymptotic behavior of the largest eigenvalues of heavy-tailed sample covariance matrices, showing that previous results hold under more general growth rates of p relative to n, using large deviation techniques.

Contribution

It proves that the convergence results for the largest eigenvalues of heavy-tailed covariance matrices hold for polynomial growth rates of p, employing new proof techniques based on large deviations.

Findings

Largest eigenvalues converge to a Fréchet distribution.

Results hold for polynomial growth of p relative to n.

Eigenvectors are close to canonical basis vectors.

Abstract

In this paper we study the joint distributional convergence of the largest eigenvalues of the sample covariance matrix of a $p$-dimensional time series with iid entries when $p$ converges to infinity together with the sample size $n$. We consider only heavy-tailed time series in the sense that the entries satisfy some regular variation condition which ensures that their fourth moment is infinite. In this case, Soshnikov [31, 32] and Auffinger et al. [2] proved the weak convergence of the point processes of the normalized eigenvalues of the sample covariance matrix towards an inhomogeneous Poisson process which implies in turn that the largest eigenvalue converges in distribution to a Fr\'echet distributed random variable. They proved these results under the assumption that $p$ and $n$ are proportional to each other. In this paper we show that the aforementioned results remain valid if $p$ grows at any polynomial rate. The proofs are different from those in [2, 31, 32]; we employ large deviation techniques to achieve them. The proofs reveal that only the diagonal of the sample covariance matrix is relevant for the asymptotic behavior of the largest eigenvalues and the corresponding eigenvectors which are close to the canonical basis vectors. We also discuss extensions of the results to sample autocovariance matrices.