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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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