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Markov chains conditioned never to wait too long at the origin

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Jacka, Saul D.. (2009) Markov chains conditioned never to wait too long at the origin. Journal of Applied Probability, Vol.46 (No.3). pp. 812-826. ISSN 0021-9002

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Official URL: http://dx.doi.org/10.1239/jap/1253279853

Abstract

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.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Markov processes, Probabilities, Branching processes, Boundary value problems, Convergence
Journal or Publication Title:	Journal of Applied Probability
Publisher:	Applied Probability Trust
ISSN:	0021-9002
Date:	September 2009
Volume:	Vol.46
Number:	No.3
Page Range:	pp. 812-826
Identification Number:	10.1239/jap/1253279853
Status:	Peer Reviewed
Access rights to Published version:	Open Access
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URI:	http://wrap.warwick.ac.uk/id/eprint/2490