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November 2019 The eigenstructure of the sample covariance matrices of high-dimensional stochastic volatility models with heavy tails
Johannes Heiny, Thomas Mikosch
Bernoulli 25(4B): 3590-3622 (November 2019). DOI: 10.3150/18-BEJ1103


We consider a $p$-dimensional time series where the dimension $p$ increases with the sample size $n$. The resulting data matrix $\mathbf{X}$ follows a stochastic volatility model: each entry consists of a positive random volatility term multiplied by an independent noise term. The volatility multipliers introduce dependence in each row and across the rows. We study the asymptotic behavior of the eigenvalues and eigenvectors of the sample covariance matrix $\mathbf{X}\mathbf{X}'$ under a regular variation assumption on the noise. In particular, we prove Poisson convergence for the point process of the centered and normalized eigenvalues and derive limit theory for functionals acting on them, such as the trace. We prove related results for stochastic volatility models with additional linear dependence structure and for stochastic volatility models where the time-varying volatility terms are extinguished with high probability when $n$ increases. We provide explicit approximations of the eigenvectors which are of a strikingly simple structure. The main tools for proving these results are large deviation theorems for heavy-tailed time series, advocating a unified approach to the study of the eigenstructure of heavy-tailed random matrices.


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Johannes Heiny. Thomas Mikosch. "The eigenstructure of the sample covariance matrices of high-dimensional stochastic volatility models with heavy tails." Bernoulli 25 (4B) 3590 - 3622, November 2019.


Received: 1 February 2018; Revised: 1 September 2018; Published: November 2019
First available in Project Euclid: 25 September 2019

zbMATH: 07110149
MathSciNet: MR4010966
Digital Object Identifier: 10.3150/18-BEJ1103

Keywords: cluster Poisson limit , convergence , dependent entries , Fréchet distribution , infinite variance stable limit , large deviations , largest eigenvalues , point process , regular variation , sample autocovariance matrix , Trace

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability


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Vol.25 • No. 4B • November 2019
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