Estimates of the supremum of a class of stationary random processes
type: Item
instance of: scholarly article
title: Estimates of the supremum of a class of stationary random processes
zbMATH Open document ID: 0758.60034
DOI: 10.1007/BF01066690
author: M. V. Ehndzhirgli
published in: Ukrainian Mathematical Journal
publication date: 1992-09-27T00:00:00Z
review text: Let \(\xi(X)\) be a stationary process with \(E\xi(X)=0\), and \(E\xi(t+\tau)\xi(t)=\int_ 0^ \infty \cos \lambda\tau dF(\lambda)\). \(F(\lambda)\) is a spectral function. Let \(F(\lambda)\) be continuous, \(F(0)=0\), \(F(\infty)=1\), \(1-F(\lambda)>0\), \(\lambda<\infty\). The process \(\xi(t)\) has a representation in the form of stochastic integral \[ \xi(t)=\int_ 0^ \infty \cos \lambda\tau d\eta_ 1(\lambda)+\int_ 0^ \infty \sin \lambda\tau d\eta_ 2(\lambda), \] where \(\eta_ i(\lambda)\) are orthogonal processes with orthogonal increments, \(E\eta_ i(\lambda)=0\), \(E| d\eta_ i(\lambda)|^ 2=dF(\lambda)\), \(i=1,2\). It is assumed that \(\eta_ i(\lambda)\in\text{Sub}_ \varphi(\Omega)\). Estimations for the distribution of the supremum of \(\xi(t)\) are obtained. Sufficient conditions are given in terms of the spectral function \(F(\lambda)\).
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 60H05
zbMATH DE Number: 58674
zbMATH Keywords: stationary process
zbMATH Keywords: stochastic integral
zbMATH Keywords: orthogonal processes with orthogonal increments
zbMATH Keywords: supremum
zbMATH Keywords: spectral function
MaRDI profile type: Publication
cites work: Estimates on the distribution of the supremum of a stationary Gaussian process
cites work: Q3763330
cites work: Q5787627
full work available at URL: https://doi.org/10.1007/bf01066690
OpenAlex ID: W2115487228
Recommended article: Estimates of the supremum distribution for a certain class of random processes
Recommended article: Estimates on the distribution of the supremum of a stationary Gaussian process
Recommended article: Q4883996
Recommended article: Exact asymptotics of supremum of a stationary Gaussian process over a random interval
Recommended article: Approximation of supremum of max-stable stationary processes \& Pickands constants
Recommended article: Distribution of the supremum location of stationary processes
Recommended article: On the supremum of certain families of stochastic processes
Recommended article: Exact asymptotics and limit theorems for supremum of stationary \(\chi\)-processes over a random interval
Recommended article: Bounds for the expected supremum of some non-stationary Gaussian processes
description: scientific article; zbMATH DE number 58674