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

TY - JOUR

T1 - Eigenvalues and eigenvectors of heavy-tailed sample covariance matrices with general growth rates: the iid case.

AU - Heiny, Johannes

AU - Mikosch, Thomas Valentin

PY - 2017/7

Y1 - 2017/7

N2 - 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 (2004, 2006) and Auffinger et al. (2009) 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 Fréchet 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 Auffinger et al. (2009) and Soshnikov (2004, 2006); 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.

AB - 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 (2004, 2006) and Auffinger et al. (2009) 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 Fréchet 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 Auffinger et al. (2009) and Soshnikov (2004, 2006); 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.

U2 - 10.1016/j.spa.2016.10.006

DO - 10.1016/j.spa.2016.10.006

M3 - Journal article

SN - 0304-4149

VL - 127

SP - 2179

EP - 2242

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

IS - 7

ER -