Q1094744
type: Item
instance of: scholarly article
title: Étude des extrêmes d'une suite stationnaire m-dépendante avec une application relative aux accroissements du processus de Wiener. (Study of the extremes of a stationary m-dependent sequence with an application to the increments of Wiener processes)
zbMATH Open document ID: 0631.60031
author: George Haiman
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77300
review text: The author investigates a stationary m-dependent sequence \(\{X_ i\}\) of random variables. In the first part, a uniform bound for \[ \mu^{- n}(u)P\{\max (X_ 1,...,X_ n)\leq n\}-1 \] is given for a certain \(\mu\) (u) and for large x with \(x<\sup \{y;P(X<y)<1\}\). This completes a previous result obtained by the author [Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. B 17, 309-330 (1981; Zbl 0479.60044)]. The result is applied to improve an evaluation of the probability that the increments of the Wiener process remain under a given threshold. In the second part the records for \(\{X_ n\}\) are considered. Let \(\{(T_ n,O_ n)\}\) be the sequence of record times and record values associated with \(\{X_ n\}\), and let \(\{(S_ n,R_ n)\}\) be the corresponding sequence associated with the sequence \(\{X_ n'\}\) of independent random variables having the same distribution as \(X_ 1\). It is shown that \(\{X_ n'\}\) can be defined so that \(S_ n=T_{n-q}\) and \(R_ n=O_{n-q}\) for an integer q and for all large n.
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 62G30
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 4026447
zbMATH Keywords: extreme value theory
zbMATH Keywords: increments of the Wiener process
zbMATH Keywords: sequence of record times
zbMATH Keywords: record values
reviewed by: Toshio Mori
MaRDI profile type: Publication
label: Étude des extrêmes d'une suite stationnaire m-dépendante avec une application relative aux accroissements du processus de Wiener. (Study of the extremes of a stationary m-dependent sequence with an application to the increments of Wiener processes)
description: scientific article; zbMATH DE number 4026447