Asymptotic Extremal Distribution for Non-Stationary, Strongly-Dependent Data
Asymptotic Extremal Distribution for Non-Stationary, Strongly-Dependent Data作者机构：Departamento Modelización Estadística de Datos e Inteligencia Artificial (MEDIA) CURE Rocha Universidad de la República Montevideo Uruguay
出 版 物：《Advances in Pure Mathematics》 (理论数学进展（英文）)
年 卷 期：2022年第12卷第8期
页 面：479-489页
学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学]
主 题：Extreme Events Strongly Dependent Data Fisher-Tippet-Gnedenko Theory
摘 要：Fisher-Tippet-Gnedenko classical theory shows that the normalized maximum of n iid random variables with distribution F belonging to a very wide class of functions, converges in law to an extremal distribution H, that is determined by the tail of F. Extensions of this theory from the iid case to stationary and weak dependent sequences are well known from the work of Leadbetter, Lindgreen and Rootzén. In this paper, we present a very simple class of random processes that runs from iid sequences to non-stationary and strongly dependent processes, and we study the asymptotic behavior of its normalized maximum. More interesting, we show that when the process is strongly dependent, the asymptotic distribution is no longer an extremal one, but a mixture of extremal distributions. We present very simple theoretical and simulated examples of this result. This provides a simple framework to asymptotic approximations of extremes values not covered by classical extremal theory and its well-known extensions.
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