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Excursions of birth and death processes, orthogonal polynomials, and continued fractions

Part of: Markov processes Convergence and divergence of infinite limiting processes

Published online by Cambridge University Press:  14 July 2016

Fabrice Guillemin  and
Didier Pinchon
Fabrice Guillemin*
Affiliation:
France Telecom
Didier Pinchon*
Affiliation:
Université Paul Sabatier
*
∗Postal address: France Telecom/CNET DAC/ARP, Technopole Anticipa, 2 Avenue Pierre Marzin, 22307 Lannion Cedex, France. Email address: Fabrice.Guillemin@cnet.francetelecom.fr.
∗∗Postal address: Laboratoire MIP, Université Paul Sabatier, 118 route de Narbonne, 31 062 Toulouse Cedex, France.
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Abstract

On the basis of the Karlin and McGregor result, which states that the transition probability functions of a birth and death process can be expressed via the introduction of an orthogonal polynomial system and a spectral measure, we investigate in this paper how the Laplace transforms and the distributions of different transient characteristics related to excursions of a birth and death process can be expressed by means of the basic orthogonal polynomial system and the spectral measure. This allows us in particular to give a probabilistic interpretation of the series introduced by Stieltjes to study the convergence of the fundamental continued fraction associated with the system. Throughout the paper, we pay special attention to the case when the birth and death process is ergodic. Under the assumption that the spectrum of the spectral measure is discrete, we show how the distributions of different random variables associated with excursions depend on the fundamental continued fraction, the orthogonal polynomial system and the spectral measure.

Keywords

Continued fractionbirth and death processspectral measureorthogonal polynomial

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 40A15: Convergence and divergence of continued fractions
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 3 , September 1999 , pp. 752 - 770
Copyright
Copyright © Applied Probability Trust 1999 

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