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1 results

  • Markov Chains Conditioned Never to Wait Too Long at the Origin

  • Part of
  • Saul Jacka
  • Journal:
    Journal of Applied Probability / Volume 46 / Issue 3 / September 2009
    Published online by Cambridge University Press:
    14 July 2016, pp. 812-826
    Print publication:
    September 2009
    • Article
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  • 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.