On the existence of uni-instantaneous Q-processes with a given finite μ-invariant measure
Brenton Gray, Phil Pollett, and Hanjun Zhang
Source: J. Appl. Probab. Volume 42, Number 3 (2005), 713-725.
Abstract
Let S be a countable set and let Q = (qij, i,j∈S) 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, j∈S) such that ∑i∈S miqij= -μmj, j≠b. We prove that there exists a Q-process P(t)=(pij(t), i,j∈ S) for which m is a μ-invariant measure, that is ∑i∈ S mipij(t)=e-μt mj, j∈S. We illustrate our results with reference to the Kolmogorov `K1' chain and a birth-death process with catastrophes and instantaneous resurrection.
Primary Subjects: 60J27
Secondary Subjects: 60J35
Keywords: Markov chain; q-matrix; birth-death process; construction theory
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jap/1127322022
Digital Object Identifier: doi:10.1239/jap/1127322022
Mathematical Reviews number (MathSciNet):
MR2157515
Zentralblatt MATH identifier:
1084.60046
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