Limiting Behavior of Maxima under Dependence

TL;DR

This paper investigates the asymptotic behavior of maxima in dependent sequences of random variables, generalizing classical extreme value theory by incorporating dependence structures and copula-based distortions.

Contribution

It introduces a general framework for the weak convergence of maxima under dependence, extending the Fisher-Tippett-Gnedenko theorem to dependent sequences with explicit limiting distributions.

Findings

Derived generalized extreme value distributions with dependence-based distortions

Established uniform convergence rates for the weak convergence

Unified existing results for exchangeable and stationary sequences

Abstract

Weak convergence of maxima of dependent sequences of identically distributed continuous random variables is studied under normalizing sequences arising as subsequences of the normalizing sequences from an associated iid sequence. This general framework allows one to derive several generalizations of the well-known Fisher-Tippett-Gnedenko theorem under conditions on the univariate marginal distribution and the dependence structure of the sequence. The limiting distributions are shown to be compositions of a generalized extreme value distribution and a distortion function which reflects the limiting behavior of the diagonal of the underlying copula. Uniform convergence rates for the weak convergence to the limiting distribution are also derived. Examples covering well-known dependence structures are provided. Several existing results, e.g. for exchangeable sequences or stationary time series, are embedded in the proposed framework.