Large sample autocovariance matrices of linear processes with heavy\n tails

2020/01/14 by Johannes Heiny, Thomas Mikosch, Heiny, Johannes +1 · 1 citation
Mathematics · #FOS: Mathematics #Morphological variations and asymmetry #Primary 60B20 #Probability (math.PR) #Random Matrices and Applications #Secondary 60F05 60F10 60G10 60G55 60G70 #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2001.05056

openalex publication_date 2020/01/14 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

We provide asymptotic theory for certain functions of the sample\nautocovariance matrices of a high-dimensional time series with infinite fourth\nmoment. The time series exhibits linear dependence across the coordinates and\nthrough time. Assuming that the dimension increases with the sample size, we\nprovide theory for the eigenvectors of the sample autocovariance matrices and\nfind explicit approximations of a simple structure, whose finite sample quality\nis illustrated for simulated data. We also obtain the limits of the normalized\neigenvalues of functions of the sample autocovariance matrices in terms of\ncluster Poisson point processes. In turn, we derive the distributional limits\nof the largest eigenvalues and functionals acting on them. In our proofs, we\nuse large deviation techniques for heavy-tailed processes, point process\ntechniques motivated by extreme value theory, and related continuous mapping\narguments.\n

Cited by

• Asymptotic analysis of high-dimensional uniformity tests under heavy-tailed alternatives

Related

• Eigenvalues and eigenvectors of heavy-tailed sample covariance matrices\n with general growth rates: the iid case
• Limit Theory for the largest eigenvalues of sample covariance matrices with heavy-tails
• Spectral analysis of linear time series in moderately high dimensions
• Spiked eigenvalues of high-dimensional sample autocovariance matrices: CLT and applications
• Eigenvalues of sample covariance matrices of non-linear processes with infinite variance