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Zbl 0681.60095
Babillot, M.
Théorie du renouvellement pour des chaînes semi-markoviennes transientes. (Renewal theory for transient semi-Markov chains). (French. English summary)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 24, No.4, 507-569 (1988). ISSN 0246-0203

A Markov chain $((X\sb n,Y\sb n))$ with state space $X\times {\bbfR}\sp d$ 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 $(X\sb n)$ (and other assumptions), the renewal kernel (potential kernel) $U=\sum\sp{\infty}\sb{n=1}P\sp n$ satisfies $$ \lim\sb{\vert y\vert \to \infty}\sqrt{\det \sigma}\vert y\vert\sb{\sigma}\sp{d-1}U f(x,y)=c\sb d\int F\quad (w,y')\pi (dw)dy' $$ for each function f and each x, where $\sigma$ is a certain positive- definite matrix, $$ \vert y\vert\sb{\sigma}=\sqrt{\sigma\sp{-1}(y)},\quad c\sb d=\Gamma ((d-2)/2)/(2\pi)\sp{d/2}, $$ and $\pi$ is the invariant distribution of $(X\sb n)$. This result generalizes the classical renewal theorems of Blackwell for ordinary renewal processes on ${\bbfR}$ and Ney and Spitzer for random walks on ${\bbfR}\sp d$.
[A.Karr]

MSC 2000:: *60K15 Markov renewal processes
60J45 Probabilistic potential theory
60K05 Renewal theory
60J50 Boundary theory (probability)

Keywords: semi-Markov process; renewal theory; ergodicity hypotheses; renewal theorems of Blackwell

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