# An Orthogonal-Polynomial Approach to First-Hitting Times of Birth–Death Processes

Journal of Theoretical Probability, Dec 2015

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 ${\mathbb {E}}[e^{sT_{ij}}]$ 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.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs10959-015-0659-z.pdf

Erik A. van Doorn. An Orthogonal-Polynomial Approach to First-Hitting Times of Birth–Death Processes, Journal of Theoretical Probability, 2017, 594-607, DOI: 10.1007/s10959-015-0659-z