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

TL;DR

This paper introduces an orthogonal-polynomial method to analyze first-hitting times in birth-death processes, providing an alternative to Dirichlet form techniques for deriving Laplace transform representations and asymptotics.

Contribution

It demonstrates that orthogonal polynomial tools can replicate and extend recent Dirichlet form results on first-hitting times in birth-death processes.

Findings

Orthogonal-polynomial approach yields Laplace transform representations.

Method reproduces asymptotic results for large states.

Provides a new perspective using associated polynomials and Markov's theorem.

Abstract

In a recent paper in the Journal of Theoretical Probability 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 $any$ 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 $associated$ $polynomials$ and $Markov's$ $theorem$.