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April 2021 Point process convergence for the off-diagonal entries of sample covariance matrices
Johannes Heiny, Thomas Mikosch, Jorge Yslas
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Ann. Appl. Probab. 31(2): 538-560 (April 2021). DOI: 10.1214/20-AAP1597

Abstract

We study point process convergence for sequences of i.i.d. random walks. The objective is to derive asymptotic theory for the extremes of these random walks. We show convergence of the maximum random walk to the Gumbel distribution under the existence of a (2+δ)th moment. We make heavy use of precise large deviation results for sums of i.i.d. random variables. As a consequence, we derive the joint convergence of the off-diagonal entries in sample covariance and correlation matrices of a high-dimensional sample whose dimension increases with the sample size. This generalizes known results on the asymptotic Gumbel property of the largest entry.

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Johannes Heiny. Thomas Mikosch. Jorge Yslas. "Point process convergence for the off-diagonal entries of sample covariance matrices." Ann. Appl. Probab. 31 (2) 538 - 560, April 2021. https://doi.org/10.1214/20-AAP1597

Information

Received: 1 July 2019; Revised: 1 January 2020; Published: April 2021
First available in Project Euclid: 1 April 2021

Digital Object Identifier: 10.1214/20-AAP1597

Subjects:
Primary: 60G70
Secondary: 60B20 , 60F10 , 60G50 , 62F05

Keywords: Extreme value theory , Gumbel distribution , maximum entry , precise large deviations , Sample covariance matrix

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 2 • April 2021
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