Point process convergence for the off-diagonal entries of sample\n covariance matrices

2020/02/18 by Johannes Heiny, Thomas Mikosch, Heiny, Johannes +3
Mathematics · #60F10 #60G50 #62F05 #FOS: Mathematics #Point processes and geometric inequalities #Primary 60G70 #Probability (math.PR) #Random Matrices and Applications #Secondary 60B20 #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2002.07771

openalex publication_date 2020/02/18 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

We study point process convergence for sequences of iid random walks. The\nobjective is to derive asymptotic theory for the extremes of these random\nwalks. We show convergence of the maximum random walk to the Gumbel\ndistribution under the existence of a (2+\δ)th moment. We make heavily\nuse of precise large deviation results for sums of iid random variables. As a\nconsequence, we derive the joint convergence of the off-diagonal entries in\nsample covariance and correlation matrices of a high-dimensional sample whose\ndimension increases with the sample size. This generalizes known results on the\nasymptotic Gumbel property of the largest entry.\n

Related

• Gumbel and Fr 'echet convergence of the maxima of independent random\n walks
• Maximum interpoint distance of high-dimensional random vectors
• Asymptotic independence of point process and Frobenius norm of a large sample covariance matrix
• Point process convergence for symmetric functions of high-dimensional random vectors
• Eigenvalues and eigenvectors of heavy-tailed sample covariance matrices\n with general growth rates: the iid case