Bernoulli

  • Bernoulli
  • Volume 25, Number 4B (2019), 3590-3622.

The eigenstructure of the sample covariance matrices of high-dimensional stochastic volatility models with heavy tails

Johannes Heiny and Thomas Mikosch

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Abstract

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.

Article information

Source
Bernoulli, Volume 25, Number 4B (2019), 3590-3622.

Dates
Received: February 2018
Revised: September 2018
First available in Project Euclid: 25 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.bj/1569398778

Digital Object Identifier
doi:10.3150/18-BEJ1103

Mathematical Reviews number (MathSciNet)
MR4010966

Zentralblatt MATH identifier
07110149

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

Citation

Heiny, Johannes; Mikosch, Thomas. The eigenstructure of the sample covariance matrices of high-dimensional stochastic volatility models with heavy tails. Bernoulli 25 (2019), no. 4B, 3590--3622. doi:10.3150/18-BEJ1103. https://projecteuclid.org/euclid.bj/1569398778


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