Markov chains conditioned never to wait too long at the origin

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.