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The recurrence and transience of two-dimensional linear birth and death processes

Published online by Cambridge University Press:  01 July 2016

J. Hutton
J. Hutton*
Affiliation:
Bucknell University
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Abstract

A two-dimensional linear birth and death process is a continuous-time Markov chain Y(·) with state space (Z+)2 which can jump from the point (n, m) to one of its four neighbors, with rates that are linear functions of n and m. Criteria are extended for determining whether such a process has a positive probability or zero probability of escaping to infinity. In the transient case considered, the projections of the imbedded Markov chain {Xn} of the successive states visited by Y(·) on a suitable pair of orthonormal vectors v and w are shown to be regularly varying sequences with index 1. Specifically, (Xn, v)∽δn and (Xn, w)∽ kn/log n for positive constants δ and k.

Keywords

BIRTH AND DEATH PROCESSESREGULARLY VARYING SEQUENCESMARTINGALES
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 3 , September 1980 , pp. 615 - 639
Copyright
Copyright © Applied Probability Trust 1980 

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