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Markov Chains Conditioned Never to Wait Too Long at the Origin

Part of: Markov processes Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  14 July 2016

Saul Jacka
Saul Jacka*
Affiliation:
University of Warwick
*
∗Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Email address: s.d.jacka@warwick.ac.uk
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Abstract

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Motivated by Feller's coin-tossing problem, we consider the problem of conditioning an irreducible Markov chain never to wait too long at 0. Denoting by τ the first time that the chain, X, waits for at least one unit of time at the origin, we consider conditioning the chain on the event (τ›T). We show that there is a weak limit as T→∞ in the cases where either the state space is finite or X is transient. We give sufficient conditions for the existence of a weak limit in other cases and show that we have vague convergence to a defective limit if the time to hit zero has a lighter tail than τ and τ is subexponential.

Keywords

Subexponential tailevanescent processFeller's coin-tossing constantshitting probabilitiesconditioned process

MSC classification

Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J50: Boundary theory 60B10: Convergence of probability measures
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 3 , September 2009 , pp. 812 - 826
Copyright
Copyright © Applied Probability Trust 2009 

Footnotes

I am grateful to an anonymous referee for many helpful suggestions on improving the presentation of this paper.

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