Compound Poisson Approximation for Multiple Runs in a Markov Chain

摘要

We consider a sequence X 1, ..., X n of r.v.'s generated by a stationary Markov chain with state space A = {0, 1, ..., r}, r ≥ 1. We study the overlapping appearances of runs of k i consecutive i's, for all i = 1, ..., r, in the sequence X 1,..., X n. We prove that the number of overlapping appearances of the above multiple runs can be approximated by a Compound Poisson r.v. with compounding distribution a mixture of geometric distributions. As an application of the previous result, we introduce a specific Multiple-failure mode reliability system with Markov dependent components, and provide lower and upper bounds for the reliability of the system.