Conditional Markov renewal theory. I. Finite and denumerable state space
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: \((\forall_ n)\) \(S_ n\geq S_{n-1}\).
scientific article; zbMATH DE number 3878133