Asymptotic theory for the sample covariance matrix of a heavy-tailed multivariate time series

Richard A. Davis; Thomas Valentin Mikosch; Olivier  Pfaffel

doi:10.1016/j.spa.2015.10.001

Asymptotic theory for the sample covariance matrix of a heavy-tailed multivariate time series

Richard A. Davis, Thomas Valentin Mikosch, Olivier Pfaffel

Department of Mathematical Sciences

7 Citations (Scopus)

Abstract

In this paper we give an asymptotic theory for the eigenvalues of the sample covariance matrix of a multivariate time series. 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 α∈(0,4) 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. 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.

Original language	English
Journal	Stochastic Processes and Their Applications
Volume	126
Issue number	3
Pages (from-to)	767–799
ISSN	0304-4149
DOIs	https://doi.org/10.1016/j.spa.2015.10.001
Publication status	Published - Mar 2016

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title = "Asymptotic theory for the sample covariance matrix of a heavy-tailed multivariate time series",

abstract = "In this paper we give an asymptotic theory for the eigenvalues of the sample covariance matrix of a multivariate time series. 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 α∈(0,4) 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. 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.",

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T1 - Asymptotic theory for the sample covariance matrix of a heavy-tailed multivariate time series

AU - Davis, Richard A.

AU - Mikosch, Thomas Valentin

AU - Pfaffel, Olivier

PY - 2016/3

Y1 - 2016/3

N2 - In this paper we give an asymptotic theory for the eigenvalues of the sample covariance matrix of a multivariate time series. 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 α∈(0,4) 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. 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.

AB - In this paper we give an asymptotic theory for the eigenvalues of the sample covariance matrix of a multivariate time series. 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 α∈(0,4) 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. 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.

U2 - 10.1016/j.spa.2015.10.001

DO - 10.1016/j.spa.2015.10.001

M3 - Journal article

SN - 0304-4149

VL - 126

SP - 767

EP - 799

JO - Stochastic Processes and Their Applications

JF - Stochastic Processes and Their Applications

IS - 3

ER -

Asymptotic theory for the sample covariance matrix of a heavy-tailed multivariate time series

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