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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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