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Renewal theory for functionals of a Markov chain with compact state space. (English) Zbl 1048.60065
The main result of the paper is the key renewal theorem for Markov chains \((x_n)_{n\geq 0}\) with compact state space \(S\): Under some conditions, such as existence of stationary distribution \(\pi(dx),\) \(\lim_{n\to+\infty}(v_n/n)=\beta,\) where \((v_n)_{n\geq 0}\) is the corresponding time changed process, the following limit exists: \[ \lim_{t\to+\infty}\sum_{k=0}^{\infty} g(x_k,t-v_k)= \frac{1}{\beta}\int_S\pi(dx) \int_{-\infty}^{+\infty}g(x,t) \,dt,\quad \forall x\in S, \] where \(g\) is continuous bounded function. This is a modification of H. Kesten’s result [Ann. Probab. 2, 355–386 (1974; Zbl 0303.60090)].

60K05 Renewal theory
60J05 Discrete-time Markov processes on general state spaces
60K15 Markov renewal processes, semi-Markov processes
60H25 Random operators and equations (aspects of stochastic analysis)
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