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Heavy-tailed asymptotics of stationary probability vectors of Markov chains of gi/g/1 type

Published online by Cambridge University Press:  01 July 2016

Quan-Lin Li*
Affiliation:
Tsinghua University
Yiqiang Q. Zhao*
Affiliation:
Carleton University
*
Postal address: Department of Industrial Engineering, Tsinghua University, Beijing 100084, P. R. China.
Postal address: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6, Canada. Email address: zhao@math.carleton.ca
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Abstract

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In this paper, we provide a novel approach to studying the heavy-tailed asymptotics of the stationary probability vector of a Markov chain of GI/G/1 type, whose transition matrix is constructed from two matrix sequences referred to as a boundary matrix sequence and a repeating matrix sequence, respectively. We first provide a necessary and sufficient condition under which the stationary probability vector is heavy tailed. Then we derive the long-tailed asymptotics of the R-measure in terms of the RG-factorization of the repeating matrix sequence, and a Wiener-Hopf equation for the boundary matrix sequence. Based on this, we are able to provide a detailed analysis of the subexponential asymptotics of the stationary probability vector.

Type
General Applied Probability
Copyright
Copyright © Applied Probability Trust 2005 

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