Conditional Markov renewal theory. I. Finite and denumerable state space
type: Item
instance of: scholarly article
title: Conditional Markov renewal theory. I. Finite and denumerable state space
zbMATH Open document ID: 0551.60094
published in: The Annals of Probability
publication date: 1984-01-01T00:00:00Z
review text: A semi-Markov process with a denumerable state space \({\mathcal Y}\) is considered. Conditional distributions given a sequence of states are investigated. Let \((S_ n,Y_ n)_ 0^{\infty}\) be a sequence of jump instants and values of the process in these instants \((S_ 0=0)\), \({\mathcal F}=\sigma (Y_ n, n\in \{0,1,2,...\})\). Under the assumption that \(P(S_ 1\in dz,\quad Y_ 1\in A| Y_ 0=y)=P(Y_ 1\in A| Y_ 0=y)F_ y(dz),\) and that the Markov chain \((Y_ n)_ 0^{\infty}\) is aperiodic, irreducible and recurrent with an invariant measure \(\pi\) (\(\cdot)\) the following properties are proved: (\(\forall y\in {\mathcal Y})\) \(P_ y\)-a.s. \((\forall h>0)(\forall A\subset {\mathcal Y})\) 1) \(U_ y(A,[a,a+h)| {\mathcal F})\to h\pi (A)\mu^{-1},\) 2)P\({}_ y(S_{\tau (a)}-a\geq h\), \(Y_{\tau (a)}\in A| {\mathcal F})\to G([h,\infty)\times A)\) (a\(\to \infty)\) where \(U_ y(B| {\mathcal F})=\sum^{\infty}_{n=1}P_ y((S_ n,Y_ n)\in B| {\mathcal F})\), \(\tau (a)=\min \{n:S_ n>a\}\), G is some probability measure on \(R_+\times {\mathcal Y}\), \(\mu =\sum_{y\in {\mathcal Y}}\mu_ y\pi (y)\), \(\mu_ y=E(S_ 1| Y_ 0=y)\). The results hold also for the sequence \((S_ n)\) without the necessary semi-Markov condition: \((\forall_ n)\) \(S_ n\geq S_{n-1}\).
reviewed by: B. P. Harlamov
Mathematics Subject Classification ID: 60K15
Mathematics Subject Classification ID: 60K05
zbMATH DE Number: 3878133
zbMATH Keywords: semi-Markov process
zbMATH Keywords: Conditional distributions
zbMATH Keywords: invariant measure
author: Steven P. Lalley
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176993144
OpenAlex ID: W2063024474
DOI: 10.1214/AOP/1176993144
Recommended article: Renewal theory for functionals of a Markov chain with compact state space.
Recommended article: A multi-dimensional renewal theorem for finite Markov chains
Recommended article: On the Theory of Markov Renewal
Recommended article: Q3842619
Recommended article: Q3333839
Recommended article: Q5830906
Recommended article: Q3788836
Recommended article: Markov Renewal and Piecewise Deterministic Processes
Recommended article: Q5816990
description: scientific article; zbMATH DE number 3878133