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Representations for the Decay Parameter of a Birth-Death Process Based on the Courant-Fischer Theorem

Part of:Markov processesNontrigonometric harmonic analysis

Published online by Cambridge University Press:  30 January 2018

Erik A. van Doorn
Erik A. van Doorn*
Affiliation:
University of Twente
*
∗ Postal address: Department of Applied Mathematics, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands. Email address: e.a.vandoorn@utwente.nl
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Abstract

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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.

Keywords

Birth-death processexponential decayrate of convergenceorthogonal polynomials

MSC classification

Primary:60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary:42C05: Orthogonal functions and polynomials, general theory

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 1 , March 2015, pp. 278 - 289
Copyright
© Applied Probability Trust 

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