## Théorie du renouvellement pour des chaînes semi-markoviennes transientes. (Renewal theory for transient semi-Markov chains).(French. English summary)Zbl 0681.60095

A Markov chain $$((X_ n,Y_ n))$$ with state space $$X\times {\mathbb{R}}^ d$$ is termed semi-Markovian if the transition kernel P satisfies $\int P((x,y+z),dx'\times dy')f(x',y')=\int P((x,y),dx'\times dy')f(x',y'+z)$ for each x,y and z. The author studies the renewal theory of such processes, showing, in particular, that under appropriate ergodicity hypotheses on the Markov chain $$(X_ n)$$ (and other assumptions), the renewal kernel (potential kernel) $$U=\sum^{\infty}_{n=1}P^ n$$ satisfies $\lim_{| y| \to \infty}\sqrt{\det \sigma}| y|_{\sigma}^{d-1}U f(x,y)=c_ d\int F\quad (w,y')\pi (dw)dy'$ for each function f and each x, where $$\sigma$$ is a certain positive- definite matrix, $| y|_{\sigma}=\sqrt{\sigma^{-1}(y)},\quad c_ d=\Gamma ((d-2)/2)/(2\pi)^{d/2},$ and $$\pi$$ is the invariant distribution of $$(X_ n)$$. This result generalizes the classical renewal theorems of Blackwell for ordinary renewal processes on $${\mathbb{R}}$$ and Ney and Spitzer for random walks on $${\mathbb{R}}^ d$$.
Reviewer: A.Karr

### MSC:

 60K15 Markov renewal processes, semi-Markov processes 60J45 Probabilistic potential theory 60K05 Renewal theory 60J50 Boundary theory for Markov processes
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