An orthogonal-polynomial approach to first-hitting times of birth-death processes

Erik A. van Doorn

Research output: Contribution to journal › Article › Academic › peer-review

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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 E[exp(sTij)] of the first-hitting time Tij for any pair of states i and j, as well as asymptotics for E[exp(sTij)] 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.
Original languageEnglish
Pages (from-to)594-607
Number of pages13
JournalJournal of Theoretical Probability
Volume30
Issue number2
DOIs
Publication statusPublished - Jun 2017

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First Hitting Time
Birth-death Process
Orthogonal Polynomials
Associated Polynomials
Dirichlet Form
Laplace transform
Non-negative
Infinity
Tend
Integer
Theorem
Polynomials
Dirichlet

Keywords

  • Orthogonal polynomials
  • Associated polynomials
  • Markov’s theorem

Cite this

van Doorn, Erik A. / An orthogonal-polynomial approach to first-hitting times of birth-death processes. In: Journal of Theoretical Probability. 2017 ; Vol. 30, No. 2. pp. 594-607.
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van Doorn, EA 2017, 'An orthogonal-polynomial approach to first-hitting times of birth-death processes' Journal of Theoretical Probability, vol. 30, no. 2, pp. 594-607. https://doi.org/10.1007/s10959-015-0659-z

An orthogonal-polynomial approach to first-hitting times of birth-death processes. / van Doorn, Erik A.

In: Journal of Theoretical Probability, Vol. 30, No. 2, 06.2017, p. 594-607.

Research output: Contribution to journal › Article › Academic › peer-review

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PY - 2017/6

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AB - 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 E[exp(sTij)] of the first-hitting time Tij for any pair of states i and j, as well as asymptotics for E[exp(sTij)] 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.

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KW - Associated polynomials

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van Doorn EA. An orthogonal-polynomial approach to first-hitting times of birth-death processes. Journal of Theoretical Probability. 2017 Jun;30(2):594-607. https://doi.org/10.1007/s10959-015-0659-z

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