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Wall and Siegmund Duality Relations for Birth and Death Chains with Reflecting Barrier

Author

Listed:
  • Holger Dette

    (Ruhr-Universität Bochum)

  • James Allen Fill

    (The Johns Hopkins University)

  • Jim Pitman

    (University of California)

  • William J. Studden

    (Purdue University)

Abstract

For a birth and death chain on the nonnegative integers with birth and death probabilities p i and q i≡ 1 –p i and reflecting barrier at 0, it is shown that the right tails of the probability of the first return from state 0 to state 0 are simple transition probabilities of a dual birth and death chain obtained by switching p iand q i. Combinatorial and analytical proofs are presented. Extensions and relations to other concepts of duality in the literature are discussed.

Suggested Citation

  • Holger Dette & James Allen Fill & Jim Pitman & William J. Studden, 1997. "Wall and Siegmund Duality Relations for Birth and Death Chains with Reflecting Barrier," Journal of Theoretical Probability, Springer, vol. 10(2), pages 349-374, April.
    • Handle: RePEc:spr:jotpro:v:10:y:1997:i:2:d:10.1023_a:1022660400024
    DOI: 10.1023/A:1022660400024
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    References listed on IDEAS

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    1. Cox, J. Theodore & Rösler, Uwe, 1984. "A duality relation for entrance and exit laws for Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 16(2), pages 141-156, February.
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    Cited by:

    1. Baake, E. & Esercito, L. & Hummel, S., 2023. "Lines of descent in a Moran model with frequency-dependent selection and mutation," Stochastic Processes and their Applications, Elsevier, vol. 160(C), pages 409-457.
    2. Thierry E. Huillet, 2022. "Chance Mechanisms Involving Sibuya Distribution and its Relatives," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(2), pages 722-764, November.
    3. Thierry E. Huillet, 2020. "On New Mechanisms Leading to Heavy-Tailed Distributions Related to the Ones Of Yule-Simon," Indian Journal of Pure and Applied Mathematics, Springer, vol. 51(1), pages 321-344, March.

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