Asymptotic Extremal Distribution for Non-Stationary, Strongly-Dependent Data  

作　　者：Carolina Crisci Gonzalo Perera Carolina Crisci;Gonzalo Perera(Departamento Modelización Estadística de Datos e Inteligencia Artificial (MEDIA), CURE, Rocha, Universidad de la República, Montevideo, Uruguay)

机构地区：[1]Departamento Modelización Estadística de Datos e Inteligencia Artificial (MEDIA), CURE, Rocha, Universidad de la República, Montevideo, Uruguay

出　　处：《Advances in Pure Mathematics》2022年第8期479-489,共11页理论数学进展（英文）

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.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.

关 键 词：Extreme Events Strongly Dependent Data Fisher-Tippet-Gnedenko Theory 

分 类 号：O171[理学—数学]