Strong theorems on the extreme values of stationary Poisson processes
type: Item
instance of: scholarly article
title: Strong theorems on the extreme values of stationary Poisson processes
zbMATH Open document ID: 0822.60025
DOI: 10.1016/0378-3758(94)00078-6
published in: Journal of Statistical Planning and Inference
publication date: 1995-10-15T00:00:00Z
review text: A number of strong limit theorems are derived for \(\max (Y_ 1, \dots, Y_ n)\), where \(Y_ 1, Y_ 2,\dots\) is an i.i.d. sequence of random variables with a Poisson-type tail behaviour. These results are used to study strong extreme value asymptotics for a sequence \(X_ i(t) = X_ i + W_ i(t)\), \(t \geq 0\), \(i = 1, 2, \dots, \) of stochastic processes, where \(X_ 1, X_ 2, \dots\) denote the points of a Poisson process in \(\mathbb{R}^ d\), and \(W_ i (t)\) be an i.i.d. sequence of \(\mathbb{R}^ d\)- valued Wiener processes. Almost sure upper and lower bounds are derived for various extreme value processes based on \(X_ i (t)\), e.g., for \(S(T) = \sup_{0 \leq t \leq T} s(t)\) with \(s(t)\) denoting the number of points \(X_ i(t)\) in the unit ball at time \(t\).
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G70
Mathematics Subject Classification ID: 60G55
zbMATH DE Number: 758087
zbMATH Keywords: strong limit theorems
zbMATH Keywords: strong extreme value asymptotics
zbMATH Keywords: Wiener processes
zbMATH Keywords: extreme value processes
author: Pál Révész
reviewed by: J. G. Steinebach
MaRDI profile type: Publication
full work available at URL: 0378-3758(94)00078-6
OpenAlex ID: W2002296916
cites work: Path properties of an infinite system of Wiener processes
description: scientific article; zbMATH DE number 758087