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Conditional Markov renewal theory. I. Finite and denumerable state space. (English) Zbl 0551.60094
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: $$(\forall_ n)$$ $$S_ n\geq S_{n-1}$$.
Reviewer: B.P.Harlamov

##### MSC:
 60K15 Markov renewal processes, semi-Markov processes 60K05 Renewal theory
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