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May 2018 The eigenvalues of the sample covariance matrix of a multivariate heavy-tailed stochastic volatility model
Anja Janssen, Thomas Mikosch, Mohsen Rezapour, Xiaolei Xie
Bernoulli 24(2): 1351-1393 (May 2018). DOI: 10.3150/16-BEJ901

Abstract

We consider a multivariate heavy-tailed stochastic volatility model and analyze the large-sample behavior of its sample covariance matrix. We study the limiting behavior of its entries in the infinite-variance case and derive results for the ordered eigenvalues and corresponding eigenvectors. Essentially, we consider two different cases where the tail behavior either stems from the i.i.d. innovations of the process or from its volatility sequence. In both cases, we make use of a large deviations technique for regularly varying time series to derive multivariate $\alpha$-stable limit distributions of the sample covariance matrix. For the case of heavy-tailed innovations, we show that the limiting behavior resembles that of completely independent observations. In contrast to this, for a heavy-tailed volatility sequence the possible limiting behavior is more diverse and allows for dependencies in the limiting distributions which are determined by the structure of the underlying volatility sequence.

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Anja Janssen. Thomas Mikosch. Mohsen Rezapour. Xiaolei Xie. "The eigenvalues of the sample covariance matrix of a multivariate heavy-tailed stochastic volatility model." Bernoulli 24 (2) 1351 - 1393, May 2018. https://doi.org/10.3150/16-BEJ901

Information

Received: 1 May 2016; Revised: 1 September 2016; Published: May 2018
First available in Project Euclid: 21 September 2017

zbMATH: 06778367
MathSciNet: MR3706796
Digital Object Identifier: 10.3150/16-BEJ901

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

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Vol.24 • No. 2 • May 2018
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