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

Erik A. van Doorn

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Abstract

In a recent paper [J. Theor. Probab. 25 (2012) 950-980] 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 $\mathbb{E}[e^{sT_{ij}}]$ of the first-hitting time $T_{ij}$ for {\em any} pair of states $i$ and $j$, as well as asymptotics for $\mathbb{E}[e^{sT_{ij}}]$ 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 {\em associated polynomials} and {\em Markov's Theorem}.
Original languageUndefined
Place of PublicationEnschede
PublisherUniversity of Twente, Department of Applied Mathematics
Number of pages13
Publication statusPublished - Apr 2015

Publication series

NameMemorandum of the Department of Applied Mathematics
No.2044
ISSN (Print)1874-4850

Keywords

  • IR-95675
  • METIS-312553
  • First-hitting time
  • EWI-25939
  • Orthogonal polynomials
  • Associated polynomials
  • Markov's Theorem
  • MSC-42C05
  • MSC-60J80
  • Birth-death process

Cite this

van Doorn, E. A. (2015). An orthogonal-polynomial approach to first-hitting times of birth-death processes. (Memorandum of the Department of Applied Mathematics; No. 2044). Enschede: University of Twente, Department of Applied Mathematics.