Compound Poisson approximation for multiple runs in a Markov chain

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 greater than or equal to 1. We study the overlapping appearances of runs of k(i) consecutive i's, for all i = 1,..., tau, 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.