Compound Poisson approximation for Markov chains using Stein's method
成果类型：
Article
署名作者：
Erhardsson, T
署名单位：
Royal Institute of Technology
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
发表日期：
1999
页码：
565-596
关键词：
摘要：
Let eta be a stationary Harris recurrent Markov chain on a Polish state space (S,F), with stationary distribution mu. Let Psi(n):= Sigma(i=1)(n) I(eta(i) is an element of S-1} be the number of visits to S-1 is an element of F by eta, where S-1 is rare in the sense that mu(S-1) is small. We want to find an approximating compound Poisson distribution for L(Psi(n)), such that the approximation error, measured using the total variation distance, can be explicitly bounded with a bound of order not much larger than mu(S-1). This is motivated by the observation that approximating Poisson distributions often give larger approximation errors when the visits to S-1 by eta tend to occur in clumps and also by the compound Poisson limit theorems of classical extreme value theory. We here propose an approximating compound Poisson distribution which in a natural way takes into account the regenerative properties of Harris recurrent Markov chains. A total variation distance error bound for this approximation is derived, using the compound Poisson Stein equation of Barbour, Chen and Loh and certain couplings. When the chain has an atom S-0 (e.g., a singleton) 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. As by-products we also get upper and lower bounds for the error in the approximation with Poisson or normal distributions. The above results are illustrated by numerical evaluations of the error bound for some Markov chains on finite state spaces.