On the distribution of first passage and return times for small sets. (English) Zbl 0591.60063

Cogburn, Robert

Ann. Probab. 13, 1219-1223 (1985).

Author’s abstract: For a Harris recurrent Markov chain with invariant initial distribution \(\pi\), we consider the return times \(\tau\), to state sets \(A_{\epsilon}\) with \(0<\pi (A_{\epsilon})\to 0\) as \(\epsilon\) \(\to 0\) and show that, provided the probability of early returns to \(A_{\epsilon}\) approaches 0, the \(\tau_{\epsilon}\), multiplied by suitable scaling factors, are asymptotically exponentially distributed.

Reviewer: M.Schäl

MSC:

60J05	Discrete-time Markov processes on general state spaces
60G10	Stationary stochastic processes
60K05	Renewal theory
60E05	Probability distributions: general theory
60F05	Central limit and other weak theorems

Keywords:

first passage times; state sets of small probability; exponential; distribution; Harris recurrent Markov chain; invariant initial distribution