Markov chains conditioned never to wait too long at the origin

2009/06/21 by Saul Jacka, Jacka, Saul
Business, Management and Accounting · Mathematics · #Advanced Queuing Theory Analysis #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #math.PR #msc:60B10 #msc:60J27 #msc:60J50 #msc:60J80

paper · pdf · doi:10.48550/arxiv.0906.3876

arxiv created 2009/06/21 · arxiv updated 2009/12/01

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 there is a weak limit as T→ ∞ in the cases where either the statespace 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.

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