Estimates of deviations of transition probabilities of inhomogeneous Markov processes
type: Item
instance of: scholarly article
title: Estimates of deviations of transition probabilities of inhomogeneous Markov processes
zbMATH Open document ID: 0675.60063
DOI: 10.1007/BF01057174
published in: Ukrainian Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: Let \(X^{(i)}=(X^{(i)}(t))\), \(i=1,2\), be two Markov processes with values in a measurable space (X,\({\mathfrak B}_ x)\). Let \(p^{(i)}(t,x,s,A)\) \((i=1,2\); \(x\in X\), \(A\in {\mathfrak B}_ x\), \(0\leq t\leq s<\infty)\) denote their transition probabilities. The author introduces the mixing coefficients \[ \rho^{(i)}(t,s)=\sup \{| p^{(i)}(t,x_ 1,s,A)-p^{(i)}(t,x_ 2,s,A)|:\quad x_ 1,x_ 2\in X,\quad A\in {\mathfrak B}_ x\},\quad 0\leq t\leq s<\infty,\text{ and the bounds} \] \[ \psi^{(1,2)}(t,s)=\sup_{x,A}| p^{(1)}(t,x,s,A)-p^{(2)}(t,x,s,A)|,\quad 0\leq t\leq s<\infty \] for the deviations of the transition probabilities of \(X^{(1)}\) and \(X^{(2)}\). Conditions in terms of \(\rho^{(1)}\) and \(\psi^{(1,2)}\) are given which are sufficient for \(\rho^{(2)}(t,s)\) to decrease exponentially fast as \(0\leq t-s\to \infty.\) As a consequence, the author shows in the case when \(X^{(1)}\) and \(X^{(2)}\) are time-homogeneous and satisfy certain conditions that \(X^{(i)}\) has a stationary distribution \(\pi^{(i)}\) \((i=1,2)\), and he obtains an upper bound for the total variation between \(\pi^{(1)}\) and \(\pi^{(2)}\). Finally, large deviation results for certain hitting times are derived.
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60J05
zbMATH DE Number: 4105994
zbMATH Keywords: transition probabilities
zbMATH Keywords: stationary distribution
zbMATH Keywords: large deviation
zbMATH Keywords: hitting times
author: V. V. Anisimov
reviewed by: Klaus Schürger
MaRDI profile type: Publication
cites work: Q3269536
cites work: Q5560061
cites work: Q3343277
cites work: Q5578160
cites work: Q4147381
cites work: Q3969669
cites work: Markov chains with almost exponential hitting times
description: scientific article; zbMATH DE number 4105994