Extreme value analysis for the sample autocovariance matrices of heavy-tailed multivariate time series

Abstract

We provide some asymptotic theory for the largest eigenvalues of a sample covariance matrix of a p-dimensional time series where the dimension p = p n converges to infinity when the sample size n increases. We give a short overview of the literature on the topic both in the light- and heavy-tailed cases when the data have finite (infinite) fourth moment, respectively. Our main focus is on the heavy-tailed case. In this case, one has a theory for the point process of the normalized eigenvalues of the sample covariance matrix in the iid case but also when rows and columns of the data are linearly dependent. We provide limit results for the weak convergence of these point processes to Poisson or cluster Poisson processes. Based on this convergence we can also derive the limit laws of various function als of the ordered eigenvalues such as the joint convergence of a finite number of the largest order statistics, the joint limit law of the largest eigenvalue and the trace, limit laws for successive ratios of ordered eigenvalues, etc. We also develop some limit theory for the singular values of the sample autocovariance matrices and their sums of squares. The theory is illustrated for simulated data and for the components of the S&P 500 stock index.

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Correspondence to Johannes Heiny.

Additional information

Richard Davis was supported by ARO MURI grant W911NF-12-1-0385. Thomas Mikosch’s and Johannes Heiny’s research is partly supported by the Danish Research Council Grant DFF-4002-00435 “Large random matrices with heavy tails and dependence”.

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Davis, R.A., Heiny, J., Mikosch, T. et al. Extreme value analysis for the sample autocovariance matrices of heavy-tailed multivariate time series. Extremes 19, 517–547 (2016). https://doi.org/10.1007/s10687-016-0251-7

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Keywords

  • Regular variation
  • Sample covariance matrix
  • Dependent entries
  • Largest eigenvalues
  • Trace
  • Point process convergence
  • Cluster poisson limit
  • Infinite variance stable limit
  • Fréchet distribution

AMS 2000 Subject Classifications

  • Primary 60B20; Secondary 60F05
  • 60F10
  • 60G10
  • 60G55
  • 60G70