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

Research output: Book/ReportReportOther research output

1 Citation (Scopus)

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 language Undefined Enschede University of Twente, Department of Applied Mathematics 13 Published - Apr 2015

Publication series

Name Memorandum of the Department of Applied Mathematics 2044 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.
van Doorn, Erik A. / An orthogonal-polynomial approach to first-hitting times of birth-death processes. Enschede : University of Twente, Department of Applied Mathematics, 2015. 13 p. (Memorandum of the Department of Applied Mathematics; 2044).
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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}.",
keywords = "IR-95675, METIS-312553, First-hitting time, EWI-25939, Orthogonal polynomials, Associated polynomials, Markov's Theorem, MSC-42C05, MSC-60J80, Birth-death process",
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series = "Memorandum of the Department of Applied Mathematics",
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van Doorn, EA 2015, An orthogonal-polynomial approach to first-hitting times of birth-death processes. Memorandum of the Department of Applied Mathematics, no. 2044, University of Twente, Department of Applied Mathematics, Enschede.
Enschede : University of Twente, Department of Applied Mathematics, 2015. 13 p. (Memorandum of the Department of Applied Mathematics; No. 2044).

Research output: Book/ReportReportOther research output

TY - BOOK

T1 - An orthogonal-polynomial approach to first-hitting times of birth-death processes

AU - van Doorn, Erik A.

N1 - The second file is the latest (online first, open access) version.

PY - 2015/4

Y1 - 2015/4

N2 - 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}.

AB - 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}.

KW - IR-95675

KW - METIS-312553

KW - First-hitting time

KW - EWI-25939

KW - Orthogonal polynomials

KW - Associated polynomials

KW - Markov's Theorem

KW - MSC-42C05

KW - MSC-60J80

KW - Birth-death process

M3 - Report

T3 - Memorandum of the Department of Applied Mathematics

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

PB - University of Twente, Department of Applied Mathematics

CY - Enschede

ER -

van Doorn EA. An orthogonal-polynomial approach to first-hitting times of birth-death processes. Enschede: University of Twente, Department of Applied Mathematics, 2015. 13 p. (Memorandum of the Department of Applied Mathematics; 2044).