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May 2000 Compound Poisson approximation for counts of rare patterns in Markov chains and extreme sojourns in birth-death chains
Torkel Erhardsson
Ann. Appl. Probab. 10(2): 573-591 (May 2000). DOI: 10.1214/aoap/1019487356

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.

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Torkel Erhardsson. "Compound Poisson approximation for counts of rare patterns in Markov chains and extreme sojourns in birth-death chains." Ann. Appl. Probab. 10 (2) 573 - 591, May 2000. https://doi.org/10.1214/aoap/1019487356

Information

Published: May 2000
First available in Project Euclid: 22 April 2002

zbMATH: 1063.60007
MathSciNet: MR1768222
Digital Object Identifier: 10.1214/aoap/1019487356

Subjects:
Primary: 60E15
Secondary: 60G70 , 60J05

Keywords: birth-death chain , compound Poisson approximation , counts of rare patterns , error bound , extreme sojourns , head run , stationary Markov chain

Rights: Copyright © 2000 Institute of Mathematical Statistics

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Vol.10 • No. 2 • May 2000
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