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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Abstract

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

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

Dates
First available in Project Euclid: 17 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.jap/1429282622

Digital Object Identifier
doi:10.1239/jap/1429282622

Mathematical Reviews number (MathSciNet)
MR3336862

Zentralblatt MATH identifier
1315.60094

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

Keywords
Birth-death process exponential decay rate of convergence orthogonal polynomials

Citation

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. https://projecteuclid.org/euclid.jap/1429282622


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