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On the existence of uni-instantaneous Q-processes with a given finite μ-invariant measure

Published online by Cambridge University Press:  14 July 2016

Brenton Gray*
Affiliation:
University of Queensland
Phil Pollett*
Affiliation:
University of Queensland
Hanjun Zhang*
Affiliation:
University of Queensland
*
Postal address: Department of Mathematics, University of Queensland, Queensland 4072, Australia.
Postal address: Department of Mathematics, University of Queensland, Queensland 4072, Australia.
Postal address: Department of Mathematics, University of Queensland, Queensland 4072, Australia.
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Abstract

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Let S be a countable set and let Q = (qij, i, jS) be a conservative q-matrix over S with a single instantaneous state b. Suppose that we are given a real number μ ≥ 0 and a strictly positive probability measure m = (mj, jS) such that ∑iSmiqij = −μmj, jb. We prove that there exists a Q-process P(t) = (pij(t), i, jS) for which m is a μ-invariant measure, that is ∑iSmipij(t) = eμtmj, jS. We illustrate our results with reference to the Kolmogorov ‘K1’ chain and a birth-death process with catastrophes and instantaneous resurrection.

Type
Research Papers
Copyright
© Applied Probability Trust 2005 

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