Compound Poisson approximation for counts of rare patterns in Markov chains and extreme sojourns in birth-death chains

Abstract

We consider the number of overlappingoccurrences up to a fixed time of one or several “rare ”patterns in a stationary finite-state Markov chain. We derive a bound for the total variation distance between the distribution of this quantity and a compound Poisson distribution, using general results on compound Poisson approximation for Markov chains by Erhardsson. If the state space is $\{0, 1\}$ and the pattern is a head run ($111 \dots 111$), the bound is completely explicit and improves on an earlier bound given by Geske, Godbole, Schaffner, Skolnick and Wallstrom. In general, the bound can be computed by solving five linear equation systems of dimension at most the number of states plus the sum of the lengths of the patterns. We also give approximations with error bounds for the distributions of the first occurrence time of a head run of fixed length and the longest head run occurringup to a fixed time. Finally, we consider the sojourn time in an “extreme” subset of the state space by a stationary birth–death chain and derive a bound for the total variation distance between the distribution of this quantity and a compound Poisson distribution.