The eigenstructure of the sample covariance matrices of high-dimensional stochastic volatility models with heavy tails
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The eigenstructure of the sample covariance matrices of high-dimensional stochastic volatility models with heavy tails. / Heiny, Johannes; Mikosch, Thomas.
In: Bernoulli, Vol. 25, No. 4 B, 2019, p. 3590-3622.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - The eigenstructure of the sample covariance matrices of high-dimensional stochastic volatility models with heavy tails
AU - Heiny, Johannes
AU - Mikosch, Thomas
PY - 2019
Y1 - 2019
N2 - We consider a p-dimensional time series where the dimension p increases with the sample size n. The resulting data matrix 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 XX 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.
AB - We consider a p-dimensional time series where the dimension p increases with the sample size n. The resulting data matrix 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 XX 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.
KW - Cluster Poisson limit
KW - Convergence
KW - Dependent entries
KW - Fréchet distribution
KW - Infinite variance stable limit
KW - Large deviations
KW - Largest eigenvalues
KW - Point process
KW - Regular variation
KW - Sample autocovariance matrix
KW - Trace
UR - http://www.scopus.com/inward/record.url?scp=85072973871&partnerID=8YFLogxK
U2 - 10.3150/18-BEJ1103
DO - 10.3150/18-BEJ1103
M3 - Journal article
AN - SCOPUS:85072973871
VL - 25
SP - 3590
EP - 3622
JO - Bernoulli
JF - Bernoulli
SN - 1350-7265
IS - 4 B
ER -
ID: 230393237