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On exact and large deviation approximation for the distribution of the longest run in a sequence of two-state Markov dependent trials

Part of: Distribution theory - Probability Markov processes

Published online by Cambridge University Press:  14 July 2016

James C. Fu ,
Liqun Wang  and
W. Y. Wendy Lou
James C. Fu*
Affiliation:
University of Manitoba
Liqun Wang*
Affiliation:
University of Manitoba
W. Y. Wendy Lou*
Affiliation:
University of Toronto
*
∗ Postal address: Department of Statistics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada.
∗ Postal address: Department of Statistics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada.
∗∗∗ Postal address: Department of Public Health Sciences, University of Toronto, Toronto, Ontario M5S 1A8, Canada.
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Abstract

Consider a sequence of outcomes from Markov dependent two-state (success-failure) trials. In this paper, the exact distributions are derived for three longest-run statistics: the longest failure run, longest success run, and the maximum of the two. The method of finite Markov chain imbedding is used to obtain these exact distributions, and their bounds and large deviation approximation are also studied. Numerical comparisons among the exact distributions, bounds, and approximations are provided to illustrate the theoretical results. With some modifications, we show that the results can be easily extended to Markov dependent multistate trials.

Keywords

Success runfailure runMarkov chain imbeddingtransition probability matrixprobability of large deviation

MSC classification

Primary: 60E05: Distributions
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 2 , June 2003 , pp. 346 - 360
Copyright
Copyright © Applied Probability Trust 2003 

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