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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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