Journal of Applied Probability

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

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We consider birth-death processes taking values in N ≡ {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 { ̃pij(t) } are said to be similar if, for all i, jN, there are constants cij such that ̃pij(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 ν such that ̃pij(t) = cijeνtpij(t) for all t ≥ 0.

Article information

J. Appl. Probab. Volume 37, Number 3 (2000), 835-849.

First available in Project Euclid: 27 February 2002

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Absorption probability chain sequence invariant vector transient behaviour transition function


Lenin, R. B.; Parthasarathy, P. R.; Scheinhardt, W. R. W.; van Doorn, E. A. Families of birth-death processes with similar time-dependent behaviour. J. Appl. Probab. 37 (2000), no. 3, 835--849. doi:10.1239/jap/1014842840.

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