Asymptotic theory for the sample covariance matrix of a heavy-tailed multivariate time series
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Asymptotic theory for the sample covariance matrix of a heavy-tailed multivariate time series. / Davis, Richard A.; Mikosch, Thomas Valentin; Pfaffel, Olivier .
In: Stochastic Processes and Their Applications, Vol. 126, No. 3, 2016, p. 767–799.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
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
Y1 - 2016
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
VL - 126
SP - 767
EP - 799
JO - Stochastic Processes and their Applications
JF - Stochastic Processes and their Applications
SN - 0304-4149
IS - 3
ER -
ID: 154755231