Abstract
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.
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