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Linear birth and death processes with killing

Published online by Cambridge University Press:  14 July 2016

Samuel Karlin  and
Simon Tavaré
Samuel Karlin*
Affiliation:
Stanford University
Simon Tavaré*
Affiliation:
Colorado State University
*
∗Postal address: Department of Mathematics, Stanford University, Stanford, CA 94305, U.S.A.
∗∗Postal address: Department of Statistics, Colorado State University, Fort Collins, CO 80523, U.S.A.
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Abstract

We analyze a class of linear birth and death processes X(t) with killing. The generator is of the form λ i = bi + θ, µi = ai, γ i = ci, where γ i is the killing rate. Then P{killed in (t, t + h) | X(t) = i} = γ ih + o(h), h ↓ 0. A variety of explicit results are found, and an example from population genetics is given.

Keywords

BIRTH-DEATH PROCESSESKILLINGPOPULATION GENETICS
Type
Research Papers
Information
Journal of Applied Probability , Volume 19 , Issue 3 , September 1982 , pp. 477 - 487
Copyright
Copyright © Applied Probability Trust 1982 

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Footnotes

Research supported in part by NIH Grant 5R01 GM10452–18 and NSF Grant MCS-24310.

References

[1] Karlin, S. and Mcgregor, J. (1957) The classification of birth and death processes. Trans. Amer. Math. Soc. 86, 366–400.Google Scholar
[2] Karlin, S. and Mcgregor, J. (1958) Linear growth birth and death processes. J. Math. Mech. 7, 643–62.Google Scholar
[3] Karlin, S. and Tavaré, S. (1981) The detection of a recessive visible gene in finite populations. Genet. Res. (Camb.) 37, 33–46.Google Scholar
[4] Kingman, J. F. C. (1963) The exponential decay of Markov transition probabilities. Proc. Lond. Math. Soc. 13, 337–358.Google Scholar
[5] Moran, P. A. P. (1962) The Statistical Processes of Evolutionary Theory. Clarendon Press, Oxford.Google Scholar
[6] Robertson, A. (1978) The time to detection of recessive visible genes in small populations. Genet. Res. (Camb.) 31, 255–264.Google Scholar