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

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 E [ e s T i j ] of the...

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Bibliographic Details
Published inJournal of theoretical probability Vol. 30; no. 2; pp. 594 - 607
Main Author van Doorn, Erik A.
Format Journal Article
LanguageEnglish
Published New York Springer US 01.06.2017
Springer Nature B.V
Subjects
Dirichlet problem
Infinity
Integers
Markov analysis
Markov processes
Mathematics
Mathematics and Statistics
Polynomials
Probability Theory and Stochastic Processes
Random walk theory
Statistics
Associated polynomials
Orthogonal polynomials
Markov’s theorem
60J80
First-hitting time
42C05
Birth–death process
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Summary: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 E [ e s T i j ] of the first-hitting time T i j for any pair of states i and j , as well as asymptotics for E [ e s T i j ] 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 .
ISSN:0894-9840
1572-9230
DOI:10.1007/s10959-015-0659-z