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Families of Birth-Death Processes with Similar Time-Dependent Behaviour

R. B. Lenin, P. R. Parthasarathy, W. R. W. Scheinhardt and E. A. Van Doorn
Journal of Applied Probability
Vol. 37, No. 3 (Sep., 2000), pp. 835-849
Stable URL: http://www.jstor.org/stable/3215617
Page Count: 15
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Families of Birth-Death Processes with Similar Time-Dependent Behaviour
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Abstract

We consider birth-death processes taking values in $\mathcal{N} \equiv \{0,\, 1, ...\}$, but allow the death rate in state 0 to be positive, so that escape from N is possible. Two such processes with transition functions {pij(t)} and {p̃ij(t)} are said to be similar if, for all i, j ∈ N, there are constants cij such that p̃ij(t) = cijpij(t) for all t ≥ 0. We determine conditions on the birth and death rates of a birth-death process for the process to be a member of a family of similar processes, and we identify the members of such a family. These issues are also resolved in the more general setting in which the two processes are called similar if there are constants cij and v such that p̃ij(t) = cij evt pij(t) for all t ≥ 0.

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