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On the Markov renewal theorem. (English) Zbl 0789.60066
Summary: Let $$(S,{\mathcal S})$$ be a measurable space with countably generated $$\sigma$$-field $${\mathcal S}$$ and $$(M_ n,X_ n)_{n \geq 0}$$ a Markov chain with state space $$S \times \mathbb{R}$$ and transition kernel $$\mathbb{P}:S \times ({\mathcal S} \otimes {\mathcal B}) \to[0,1]$$. Then $$(M_ n,S_ n)_{n \geq 0}$$, where $$S_ n=X_ 0+\cdots+X_ n$$ for $$n \geq 0$$, is called the associated Markov random walk. Markov renewal theory deals with the asymptotic behavior of suitable functionals of $$(M_ n,S_ n)_{n \geq 0}$$ like the Markov renewal measure $$\sum_{n \geq 0} P((M_ n,S_ n) \in A \times(t+B))$$ as $$t \to \infty$$ where $$A \in {\mathcal S}$$ and $$B$$ denotes a Borel subset of $$\mathbb{R}$$. It is shown that the Markov renewal theorem as well as a related ergodic theorem for semi-Markov processes hold true if only Harris recurrence of $$(M_ n)_{n \geq 0}$$ is assumed. This was proved by purely analytical methods by V. M. Shurenkov [Theory Probab. Appl. 29, 247-265 (1985); translation from Teor. Veroyatn. Primen. 29, No. 2, 248-263 (1984; Zbl 0544.60082)] in the one- sided case where $$\mathbb{P}(x,S \times [0, \infty))=1$$ for all $$x \in S$$. Our proof uses probabilistic arguments, notably the construction of regeneration epochs for $$(M_ n)_{n \geq 0}$$ such that $$(M_ n,X_ n)_{n \geq 0}$$ is at least nearly regenerative and an extension of Blackwell’s renewal theorem to certain random walks with stationary, 1- dependent increments.

##### MSC:
 60K15 Markov renewal processes, semi-Markov processes 60G50 Sums of independent random variables; random walks 60K05 Renewal theory 60J05 Discrete-time Markov processes on general state spaces
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