Markov chains conditioned never to wait too long at the origin
Jacka Saul
Source: J. Appl. Probab. Volume 46, Number 3 (2009), 812-826.
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.
Primary Subjects: 60J27
Secondary Subjects: 60J80, 60J50, 60B10
Keywords: Subexponential tail; evanescent process; Feller's coin-tossing constants; hitting probabilities; conditioned process
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jap/1253279853
Digital Object Identifier: doi:10.1239/jap/1253279853
Zentralblatt MATH identifier:
05611419
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