Compound Poisson Approximation for Markov Chains using Stein’s Method

Erhardsson, Torkel

Ann. Probab., Tome 27 (1999) no. 1, p. 565-596 / Harvested from Project Euclid

Résumé

Let $\eta$ be a stationary Harris recurrent Markov chain on a Polish state space $(S, \mathscr{F})$, with stationary distribution $\mu$. Let $\Psi_n:=\sum_{i-1}^n I\{\mu\in S_1\}$ be the number of visits to $S_1\in\mathscr{F}$ 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$ \mathscr{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.

Détail

Publié le : 1999-01-14
Classification:  Compound Poisson,  approximation,  error bound,  stationary Markov chain,  Harris recurrence,  regenerative,  “rare” set,  Stein equation,  coupling,  hitting probabilities,  expected hitting times,  60E15,  60J05,  60G70