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

Erik A. van Doorn

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