COMPOUND POISSON APPROXIMATION FOR MUOTIPLE RUNS IN A MARKOV CHAIN

OURANIA CHRYSSAPHINOU AND EUTICHIA VAGGELATOU

Department of Mathematics, University of Athens, Panepistimiopolis, 15784 Athens, Greece

(Received March 27, 2000; revised October 30, 2000)

Abstract. 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.

Key words and phrases: Multiple runs, Stein-Chen method, Kolmogorov distance, Compound Poisson approximation, consecutive-k_{1}, …, k_{r}-out-of-n: MFM system.