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Compound Poisson approximation for Markov chains using Stein’s method. (English) Zbl 0942.60007

Let \(\eta\) be a stationary Harris recurrent Markov chain on a Polish state space \((S,{\mathcal F})\) with stationary distribution \(\mu\). Let \(\Psi_n:= \sum^n_{i=1} I\{\eta_i\in S_1\}\) be the number of visits to \(S_1\in{\mathcal F}\) by \(\eta\), where \(S_1\) is rare, in the sense that \(\mu(S_1)\) is “small”, the distribution of \(\Psi_n\) is approximated in total variation by a compound Poisson distribution, in a natural way which takes into account the regenerative properties of Harris recurrent Markov chains. When the chain has an atom \(S_0\) such that \(\mu(S_0)> 0\), the bound depends only on much studied quantities like hitting probabilities and expected hitting times, which satisfy Poisson’s equation. The results are illustrated by numerical evaluations of the error bound for some Markov chains on finite state spaces.

MSC:

60E15 Inequalities; stochastic orderings
60J05 Discrete-time Markov processes on general state spaces
60F05 Central limit and other weak theorems
62E17 Approximations to statistical distributions (nonasymptotic)
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