AISM 54, 411-424

© 2002 ISM

## Compound Poisson approximation for multiple 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 $X_{1},\ldots,X_{n}$ of r.v.'s generated by a stationary
Markov chain with state space ${\cal A}=\{0,1,\ldots,r\}$, $r\geq 1$. We
study the *overlapping* appearances of runs of $k_{i}$ consecutive
$i$'s, for all $i=1,\ldots,r$, in the sequence $X_{1},\ldots,X_{n}$. 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-$k_{1},\ldots,k_{r}$-out-of-$n$: MFM system.

**Source**
(TeX , DVI )