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An Orthogonal-Polynomial Approach to First-Hitting Times of Birth–Death Processes

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  • Erik A. Doorn

    (University of Twente)

Abstract

In a recent paper in this journal, Gong, Mao and Zhang, using the theory of Dirichlet forms, extended Karlin and McGregor’s classical results on first-hitting times of a birth–death process on the nonnegative integers by establishing a representation for the Laplace transform $${\mathbb {E}}[e^{sT_{ij}}]$$ E [ e s T i j ] of the first-hitting time $$T_{ij}$$ T i j for any pair of states i and j, as well as asymptotics for $${\mathbb {E}}[e^{sT_{ij}}]$$ E [ e s T i j ] when either i or j tends to infinity. It will be shown here that these results may also be obtained by employing tools from the orthogonal-polynomial toolbox used by Karlin and McGregor, in particular associated polynomials and Markov’s theorem.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:30:y:2017:i:2:d:10.1007_s10959-015-0659-z
    DOI: 10.1007/s10959-015-0659-z
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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. 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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    Cited by:

    1. Giorno, Virginia & Nobile, Amelia G., 2022. "On some integral equations for the evaluation of first-passage-time densities of time-inhomogeneous birth-death processes," Applied Mathematics and Computation, Elsevier, vol. 422(C).

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