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Stochastic monotonicity of birth–death processes

Published online by Cambridge University Press:  01 July 2016

Erik A. Van Doorn
Erik A. Van Doorn*
Affiliation:
Twente University of Technology
*
∗Postal address: Department of Applied Mathematics, Twente University of Technology, P.O. Box 217, 7500 AE Enschede, The Netherlands.
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Abstract

A birth–death process {x(t): t ≥ 0} with state space the set of non-negative integers is said to be stochastically increasing (decreasing) on the interval (t1, t2) if Pr {x(t) > i} is increasing (decreasing) with t on (t1, t2) for all i = 0, 1, 2, ···. We study the problem of finding a necessary and sufficient condition for a birth–death process with general initial state probabilities to be stochastically monotone on an interval. Concrete results are obtained when the initial distribution vector of the process is a unit vector. Fundamental in the analysis, and of independent interest, is the concept of dual birth–death processes.

Keywords

BIRTH-DEATH PROCESSESBIRTH-DEATH POLYNOMIALSDUAL BIRTH-DEATH PROCESSESKARLIN-MCGREGOR REPRESENTATIONSTOCHASTIC MONOTONICITY
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 1 , March 1980 , pp. 59 - 80
Copyright
Copyright © Applied Probability Trust 1980 

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