2016/08/24 by Johannes Heiny, Thomas Mikosch, Heiny, Johannes +1 · 1 citation
Computer Science · Mathematics · #60F10 #60G10 #60G55 #60G70 #Bayesian Methods and Mixture Models #FOS: Mathematics #Morphological variations and asymmetry #Primary 60B20 #Probability (math.PR) #Random Matrices and Applications #Secondary 60F05 #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1608.06977
openalex publication_date 2016/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study the joint distributional convergence of the largest\neigenvalues of the sample covariance matrix of a p-dimensional time series\nwith iid entries when p converges to infinity together with the sample size\nn. We consider only heavy-tailed time series in the sense that the entries\nsatisfy some regular variation condition which ensures that their fourth moment\nis infinite. In this case, Soshnikov [31, 32] and Auffinger et al. [2] proved\nthe weak convergence of the point processes of the normalized eigenvalues of\nthe sample covariance matrix towards an inhomogeneous Poisson process which\nimplies in turn that the largest eigenvalue converges in distribution to a\nFr 'echet distributed random variable. They proved these results under the\nassumption that p and n are proportional to each other. In this paper we\nshow that the aforementioned results remain valid if p grows at any\npolynomial rate. The proofs are different from those in [2, 31, 32]; we employ\nlarge deviation techniques to achieve them. The proofs reveal that only the\ndiagonal of the sample covariance matrix is relevant for the asymptotic\nbehavior of the largest eigenvalues and the corresponding eigenvectors which\nare close to the canonical basis vectors. We also discuss extensions of the\nresults to sample autocovariance matrices.\n