Journal of Applied Probability

Representations for the decay parameter of a birth-death process based on the Courant-Fischer theorem

Erik A. van Doorn

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We study the decay parameter (the rate of convergence of the transition probabilities) of a birth-death process on {0, 1, . . .}, which we allow to evanesce by escape, via state 0, to an absorbing state -1. Our main results are representations for the decay parameter under four different scenarios, derived from a unified perspective involving the orthogonal polynomials appearing in Karlin and McGregor's representation for the transition probabilities of a birth-death process, and the Courant-Fischer theorem on eigenvalues of a symmetric matrix. We also show how the representations readily yield some upper and lower bounds that have appeared in the literature.

Article information

J. Appl. Probab., Volume 52, Number 1 (2015), 278-289.

First available in Project Euclid: 17 April 2015

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 42C05: Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]

Birth-death process exponential decay rate of convergence orthogonal polynomials


van Doorn, Erik A. Representations for the decay parameter of a birth-death process based on the Courant-Fischer theorem. J. Appl. Probab. 52 (2015), no. 1, 278--289. doi:10.1239/jap/1429282622.

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