Markov chains conditioned never to wait too long at the origin

Details Markov chains conditioned never to wait too long at the origin Markov chains conditioned never to wait too long at the origin

2009-06-21

arxiv.org/abs/0906.3876v1

Mathematics

Probability

Scientific paper

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 $\tau$ 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 $(\tau>T)$. We show there is a weak limit as $T\to \infty$ 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 $\tau$ and $\tau$ is subexponential.

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