The Annals of Applied Probability

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

Torkel Erhardsson

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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.

Article information

Ann. Appl. Probab., Volume 10, Number 2 (2000), 573-591.

First available in Project Euclid: 22 April 2002

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Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60J05: Discrete-time Markov processes on general state spaces 60G70: Extreme value theory; extremal processes

Compound Poisson approximation error bound stationary Markov chain counts of rare patterns head run extreme sojourns birth-death chain


Erhardsson, Torkel. Compound Poisson approximation for counts of rare patterns in Markov chains and extreme sojourns in birth-death chains. Ann. Appl. Probab. 10 (2000), no. 2, 573--591. doi:10.1214/aoap/1019487356.

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