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Approximating kth-order two-state Markov chains

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  14 July 2016

Y. H. Wang
Y. H. Wang*
Affiliation:
Concordia University
*
∗ Postal address: Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Boulevard W., Montreal, Quebec, Canada H3G 1M8.
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Abstract

In this paper, we consider kth-order two-state Markov chains {Xi} with stationary transition probabilities. For k = 1, we construct in detail an upper bound for the total variation d(Sn, Y) = Σx |𝐏(Sn = x) − 𝐏(Y = x)|, where Sn = X1 + · ··+ Xn and Y is a compound Poisson random variable. We also show that, under certain conditions, d(Sn, Y) converges to 0 as n tends to ∞. For k = 2, the corresponding results are given without derivation. For general k ≧ 3, a conjecture is proposed.

Keywords

MARKOV DEPENDENCEBERNOULLISUM OF RANDOM VARIABLESCOMPOUND POISSONTOTAL VARIATION METRICLIMITING DISTRIBUTION

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 4 , December 1992 , pp. 861 - 868
Copyright
Copyright © Applied Probability Trust 1992 

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References

Brainerd, B. (1983) A limiting distribution for Bernoulli trials with second-order Markov dependence. J. Appl. Prob. 20, 419–422.CrossRefGoogle Scholar
Brainerd, B. and Chang, S. M. (1982) Number of occurrences in two-state Markov chains, with an application in linguistics. Canad. J. Statist. 10, 225–231.CrossRefGoogle Scholar
Bühler, W. J. (1989) Probabilistic derivation for the limit of the Markov binomial distribution. Math. Scientist. 14, 117–119.Google Scholar
Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Methuen, London.Google Scholar
Dobrušin, R. L. (1953) Limit theorems for a Markov chain of two states. Izv. Akad. Nauk. U.S.S.R. Ser. Mat. 17, 291–330.Google Scholar
Gani, J. (1982) On the probability generating function of the sum of Markov Bernoulli random variables. J. Appl. Prob. 19A, 321–326.CrossRefGoogle Scholar
Koopman, B. O. (1950) A generalization of Poisson's distribution for Markoff chains. Proc. Nat. Acad. Sci. U.S.A. 36, 202–207.CrossRefGoogle ScholarPubMed
Pedler, P. J. (1978) The Occupation Time, Number of Transitions and Waiting Time for Two-State Markov Chains. , School of Mathematical Sciences, The Flinders University of South Australia.Google Scholar
Wang, Y. H. (1981) On the limit of the Markov binomial distribution. J. Appl. Prob. 18, 937–942.CrossRefGoogle Scholar
Wang, Y. H. (1989) From Poisson to compound Poisson approximations. Math. Scientist 14, 38–49.Google Scholar
Wang, Y. H. and Bühler, W. J. (1991) Renewal process proof for the limit of the Markov binomial distribution. Math. Scientist 16, 66–68.Google Scholar