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Markov chains conditioned never to wait too long at the origin

By Saul D. Jacka

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

Topics: QA
Publisher: Applied Probability Trust
Year: 2009
OAI identifier: oai:wrap.warwick.ac.uk:2490

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Citations

  1. (1995). A ratio limit theorem for (sub) Markov chains on f1;2;::g with bounded jumps. doi
  2. (1971). An Introduction To Probability Theory and its Application Vol II". 2nd Edition.
  3. (1970). An Introduction To Probability Theory and its Applications Vol I", 3rd Edn. (revised printing).
  4. (1999). Appendix: A primer on heavy-tailed distributions
  5. (1979). Diusions, Markov processes, and martingales: Vol. I".
  6. (2002). Examples of convergence and non-convergence of Markov chains conditioned not to die'. doi
  7. (1995). Existence of quasi stationary distributions. A renewal dynamical approach. doi
  8. (2006). Kyprianou and Zbigniew Palmowski: Quasi-stationary distributions for L evy processes. doi
  9. (1981). Matrices and Markov Chains", doi
  10. (2005). On L evy processes conditioned to stay positive doi
  11. (1966). On quasi-stationary distributions in discrete time Markov chains with a denumerable in of states. doi
  12. (1997). Pollett: Non-explosivity of limits of conditioned birth and death processes. doi
  13. (1994). Roberts: Strong forms of weak convergence. doi
  14. (1995). Roberts: Weak convergence of conditioned processes on a countable state space. doi
  15. (1996). Some asymptotic results for transient random walks. doi
  16. (1997). Spitzer's condition for random walks and L evy processes. doi
  17. (1994). Weak convergence of conditioned birth and death processes doi
  18. (2005). Zorana Lazic and Jon Warren: Conditioning an additive functional of a Markov chain to stay non-negative I: survival for a long time. doi
  19. (2005). Zorana Lazic and Jon Warren: Conditioning an additive functional of a Markov chain to stay non-negative II: hitting a high level. doi

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