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November 2021 Extreme eigenvalue statistics of m-dependent heavy-tailed matrices
Bojan Basrak, Yeonok Cho, Johannes Heiny, Paul Jung
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Ann. Inst. H. Poincaré Probab. Statist. 57(4): 2100-2127 (November 2021). DOI: 10.1214/21-AIHP1152

Abstract

We analyze the largest eigenvalue statistics of m-dependent heavy-tailed Wigner matrices as well as the associated sample covariance matrices having entry-wise regularly varying tail distributions with parameter α(0,4). Our analysis extends results in the previous literature for the corresponding random matrices with independent entries above the diagonal, by allowing for m-dependence between the entries of a given matrix. We prove that the limiting point process of extreme eigenvalues is a Poisson cluster process.

Nous analysons les plus grandes valeurs propres d’une matrice de Wigner avec entrées m-dépendantes et à queue lourde, de même que pour une matrice de covariance associée avec entrées de variation régulière de paramètre α(0,4). Notre analyse étend les résultats existants pour ces matrices aléatoires avec entrées indépendantes à des entrées m-dépendantes. Nous prouvons que le processus ponctuel limite des plus grandes valeurs propres est un processus de Poisson groupé.

Acknowledgements

The work of B. Basrak has been supported in part by the SNSF/HRZZ Grant CSRP 2018-01-180549. The work of Y. Cho and P. Jung was funded in part by the National Research Foundation of Korea grant NRF-2017R1A2B2001952. J. Heiny was supported by the Deutsche Forschungsgemeinschaft (DFG) via RTG 2131 High-dimensional Phenomena in Probability – Fluctuations and Discontinuity. J.H. and B.B. thank the Mathematics Department at KAIST for the hospitality. The authors are grateful to an anonymous referee for detailed and constructive feedback.

Citation

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Bojan Basrak. Yeonok Cho. Johannes Heiny. Paul Jung. "Extreme eigenvalue statistics of m-dependent heavy-tailed matrices." Ann. Inst. H. Poincaré Probab. Statist. 57 (4) 2100 - 2127, November 2021. https://doi.org/10.1214/21-AIHP1152

Information

Received: 12 March 2020; Revised: 26 January 2021; Accepted: 1 February 2021; Published: November 2021
First available in Project Euclid: 20 October 2021

Digital Object Identifier: 10.1214/21-AIHP1152

Subjects:
Primary: 60B20
Secondary: 60F05 , 60F10 , 60G10 , 60G55 , 60G70

Keywords: Dependent random matrices , Heavy-tailed random matrices , Largest eigenvalue , marked Poisson process , Poisson cluster process , regular variation , Sample covariance matrix , Wigner matrix

Rights: Copyright © 2021 Association des Publications de l’Institut Henri Poincaré

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Vol.57 • No. 4 • November 2021
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