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On Hitting Times and Fastest Strong Stationary Times for Skip-Free and More General Chains

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  • James Allen Fill

    (The Johns Hopkins University)

Abstract

An (upward) skip-free Markov chain with the set of nonnegative integers as state space is a chain for which upward jumps may be only of unit size; there is no restriction on downward jumps. In a 1987 paper, Brown and Shao determined, for an irreducible continuous-time skip-free chain and any d, the passage time distribution from state 0 to state d. When the nonzero eigenvalues ν j of the generator on {0,…,d}, with d made absorbing, are all real, their result states that the passage time is distributed as the sum of d independent exponential random variables with rates ν j . We give another proof of their theorem. In the case of birth-and-death chains, our proof leads to an explicit representation of the passage time as a sum of independent exponential random variables. Diaconis and Miclo recently obtained the first such representation, but our construction is much simpler. We obtain similar (and new) results for a fastest strong stationary time T of an ergodic continuous-time skip-free chain with stochastically monotone time-reversal started in state 0, and we also obtain discrete-time analogs of all our results. In the paper’s final section we present extensions of our results to more general chains.

Suggested Citation

  • James Allen Fill, 2009. "On Hitting Times and Fastest Strong Stationary Times for Skip-Free and More General Chains," Journal of Theoretical Probability, Springer, vol. 22(3), pages 587-600, September.
    • Handle: RePEc:spr:jotpro:v:22:y:2009:i:3:d:10.1007_s10959-009-0233-7
    DOI: 10.1007/s10959-009-0233-7
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    References listed on IDEAS

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    1. Persi Diaconis & Laurent Miclo, 2009. "On Times to Quasi-stationarity for Birth and Death Processes," Journal of Theoretical Probability, Springer, vol. 22(3), pages 558-586, September.
    2. James Allen Fill, 2009. "The Passage Time Distribution for a Birth-and-Death Chain: Strong Stationary Duality Gives a First Stochastic Proof," Journal of Theoretical Probability, Springer, vol. 22(3), pages 543-557, September.
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    Cited by:

    1. Miclo, Laurent & Arnaudon, Marc & Coulibaly-Pasquier, Koléhè, 2024. "On Markov intertwining relations and primal conditioning," TSE Working Papers 24-1509, Toulouse School of Economics (TSE).
    2. Michael C. H. Choi & Lu-Jing Huang, 2020. "On Hitting Time, Mixing Time and Geometric Interpretations of Metropolis–Hastings Reversiblizations," Journal of Theoretical Probability, Springer, vol. 33(2), pages 1144-1163, June.
    3. L. Avena & A. Gaudillière, 2018. "Two Applications of Random Spanning Forests," Journal of Theoretical Probability, Springer, vol. 31(4), pages 1975-2004, December.
    4. Yong-Hua Mao & Feng Wang & Xian-Yuan Wu, 2015. "Large Deviation Behavior for the Longest Head Run in an IID Bernoulli Sequence," Journal of Theoretical Probability, Springer, vol. 28(1), pages 259-268, March.
    5. Laurent Miclo & Pierre Patie, 2021. "On interweaving relations," Post-Print hal-03159496, HAL.
    6. James Allen Fill, 2009. "The Passage Time Distribution for a Birth-and-Death Chain: Strong Stationary Duality Gives a First Stochastic Proof," Journal of Theoretical Probability, Springer, vol. 22(3), pages 543-557, September.
    7. James Allen Fill & Vince Lyzinski, 2014. "Hitting Times and Interlacing Eigenvalues: A Stochastic Approach Using Intertwinings," Journal of Theoretical Probability, Springer, vol. 27(3), pages 954-981, September.
    8. Chen Jia, 2019. "Sharp Moderate Maximal Inequalities for Upward Skip-Free Markov Chains," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1382-1398, September.
    9. Persi Diaconis & Laurent Miclo, 2009. "On Times to Quasi-stationarity for Birth and Death Processes," Journal of Theoretical Probability, Springer, vol. 22(3), pages 558-586, September.
    10. Yu Gong & Yong-Hua Mao & Chi Zhang, 2012. "Hitting Time Distributions for Denumerable Birth and Death Processes," Journal of Theoretical Probability, Springer, vol. 25(4), pages 950-980, December.

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    2. James Allen Fill & Vince Lyzinski, 2014. "Hitting Times and Interlacing Eigenvalues: A Stochastic Approach Using Intertwinings," Journal of Theoretical Probability, Springer, vol. 27(3), pages 954-981, September.
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