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On the age distribution of a Markov chain

Published online by Cambridge University Press:  14 July 2016

Anthony G. Pakes
Anthony G. Pakes*
Affiliation:
Monash University, Clayton, Victoria
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Abstract

This paper develops the notion of the limiting age of an absorbing Markov chain, conditional on the present state. Chains with a single absorbing state {0} are considered and with such a chain can be associated a return chain, obtained by restarting the original chain at a fixed state after each absorption. The limiting age, A(j), is the weak limit of the time given Xn = j (n → ∞).

A criterion for the existence of this limit is given and this is shown to be fulfilled in the case of the return chains constructed from the Galton–Watson process and the left-continuous random walk. Limit theorems for A (J) (J → ∞) are given for these examples.

Keywords

MARKOV CHAINRETURN CHAINSTRONG RATIO LIMIT PROPERTYR-CLASSIFICATIONGALTON—WATSON PROCESSLEFT-CONTINUOUS RANDOM WALKLIMIT THEOREMSLOCAL LIMIT THEOREMS
Type
Research Papers
Information
Journal of Applied Probability , Volume 15 , Issue 1 , March 1978 , pp. 65 - 77
Copyright
Copyright © Applied Probability Trust 1978 

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