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On Times to Quasi-stationarity for Birth and Death Processes

Author

Listed:
  • Persi Diaconis

    (Stanford University
    Université de Nice-Sophia Antipolis)

  • Laurent Miclo

    (Université de Provence
    C.N.R.S.)

Abstract

The purpose of this paper is to present a probabilistic proof of the well-known result stating that the time needed by a continuous-time finite birth and death process for going from the left end to the right end of its state space is a sum of independent exponential variables whose parameters are the negatives of the eigenvalues of the underlying generator when the right end is treated as an absorbing state. The exponential variables appear as fastest strong quasi-stationary times for successive dual processes associated to the original absorbed process. As an aftermath, we get an interesting probabilistic representation of the time marginal laws of the process in terms of “local equilibria.”

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:22:y:2009:i:3:d:10.1007_s10959-009-0234-6
    DOI: 10.1007/s10959-009-0234-6
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    References listed on IDEAS

    as
    1. 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.
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    Cited by:

    1. 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.
    2. 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.
    3. Erik A. Doorn, 2017. "An Orthogonal-Polynomial Approach to First-Hitting Times of Birth–Death Processes," Journal of Theoretical Probability, Springer, vol. 30(2), pages 594-607, June.
    4. 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.
    5. Marc Arnaudon & Koléhè Coulibaly-Pasquier & Laurent Miclo, 2024. "On Markov Intertwining Relations and Primal Conditioning," Journal of Theoretical Probability, Springer, vol. 37(3), pages 2425-2456, September.
    6. 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).
    7. James Allen Fill & Vince Lyzinski, 2016. "Strong Stationary Duality for Diffusion Processes," Journal of Theoretical Probability, Springer, vol. 29(4), pages 1298-1338, December.
    8. 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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