On the distribution of tail array sums for strongly mixing stationary sequences
type: Item
instance of: scholarly article
title: On the distribution of tail array sums for strongly mixing stationary sequences
zbMATH Open document ID: 0939.60007
author: Holger Rootzén
author: M. Ross Leadbetter
author: Laurens De Haan
published in: The Annals of Applied Probability
publication date: 1999-11-23T00:00:00Z
review text: This paper studies the asymptotic distributions of ``tail array'' sums of the form \(\sum\psi_n(X_i-u_n)\) where \(\{X_i\}\) is a strongly mixing stationary sequence, \(\psi_n\) are real functions which are constants for negative arguments, \(\psi_n(x)=\psi_n(X_+)\), and \(\{u_n\}\) are levels with \(u_n \to\infty\). The random variables \(\sum 1_{(X_i-u_n>0)}\) and \(\sum(X_i-u_n)1_{( X_i-u_n>0)}\) are simple but important examples. Two cases of the asymptotic behaviour of \(\sum\psi_n(X_i-u_n)\) are considered: (1) compound Poisson convergence \((u_n\to\infty\) at a rapid rate defined by \(n(1-F(u_n))\to\tau< \infty)\) and (2) normal convergence \((u_n\to\infty\) more slowly, with \(n(1-F(u_n))\to\infty)\). A more detailed account is given for case (2). A final main result of this paper provides a simple approach to the verification of the Lindeberg condition for the important special cases of (approximately) exponential or (by a logarithmic transformation) regularly varying tails.
reviewed by: Wiesław Dziubdziela
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G70
Mathematics Subject Classification ID: 60G10
zbMATH DE Number: 1319836
zbMATH Keywords: central limit theorem
zbMATH Keywords: compound Poisson convergence
zbMATH Keywords: damage modeling
zbMATH Keywords: tail inference
zbMATH Keywords: extreme values
zbMATH Keywords: mixing
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1028903454
OpenAlex ID: W1966533147
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DOI: 10.1214/AOAP/1028903454
description: scientific article; zbMATH DE number 1319836