Abstract
This paper is concerned with the limit theory of the extreme order statistics derived from random walks. We establish the joint convergence of the order statistics near the minimum of a random walk in terms of the Feller chains. Detailed descriptions of the limit process are given in the case of simple symmetric walks and Gaussian walks.
Cet article traite de la théorie limite des statistiques d’ordre extrêmes provenant des marches aléatoires. Nous établissons la convergence conjointe des statistiques d’ordre près du minimum d’une marche aléatoire en termes des chaînes de Feller. Des descriptions détaillées du processus limite sont données dans le cas de marches simples symétriques et des marches gaussiennes.
Funding Statement
Wenpin Tang gratefully acknowledges financial support through an NSF grant DMS-2113779 and through a start-up grant at Columbia University.
Acknowledgements
We thank G. Schehr for stimulating discussions at the early stage of this work. We also thank two anonymous referees for numerous suggestions which improved the final version of this article.
Citation
Jim Pitman. Wenpin Tang. "Extreme order statistics of random walks." Ann. Inst. H. Poincaré Probab. Statist. 59 (1) 97 - 116, February 2023. https://doi.org/10.1214/22-AIHP1254
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