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A note on the extinction rates of some birth–death processes

Published online by Cambridge University Press:  14 July 2016

Maury Bramson  and
David Griffeath
Maury Bramson*
Affiliation:
University of Minnesota–Minneapolis
David Griffeath*
Affiliation:
University of Wisconsin–Madison
*
∗Postal address: School of Mathematics, University of Minnesota–Minneapolis, Minneapolis MN 55455, U.S.A.
∗∗Postal address: Mathematics Department, University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Drive, Madison WI 53706, U.S.A.
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Abstract

In this note we are concerned with the rate of extinction of certain continuous-time birth-death processes on the positive integers with absorption at 0. The class we deal with includes birth-death processes with mean holding time h(i) at i such that h (i)∼ i–α as i →∞, 0 ≦ α< 1. In general, our result estimates to within a constant multiple the probability of non-extinction by time t. For h(i)∼ i–α, the result states that the probability of non-extinction is of order t−1/(2-α) We give an application to interacting particle systems.

Keywords

BIRTH–DEATH PROCESSESEXTINCTION RATES
Type
Short Communications
Information
Journal of Applied Probability , Volume 16 , Issue 4 , December 1979 , pp. 897 - 902
Copyright
Copyright © Applied Probability Trust 1979 

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Footnotes

Research partially supported by the National Science Foundation under grants MCS 76–07039 and MCS 78–01241.

References

[1] Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer–Verlag, Berlin.Google Scholar
[2] Feller, W. (1968) An Introduction to Probability Theory and its Applications , Vol. 1, 3rd edn. Wiley, New York.Google Scholar
[3] Griffeath, D. (1978) Annihilating and coalescing random walks on Zd. Z. Wahrscheinlichkeitsth. 46, 55–65.Google Scholar
[4] Griffeath, D. (1979) Additive and Cancellative Interacting Particle Systems. Lecture Notes in Mathematics 724, Springer–Verlag, Berlin.Google Scholar
[5] Kelly, F. P. (1977) The asymptotic behavior of an invasion process. J. Appl. Prob. 14, 584–590.Google Scholar
[6] Sawyer, S. (1979) A limit theorem for patch sizes in a selectively-neutral migration model. J. Appl. Prob. 16, 482–495.Google Scholar
[7] Karlin, S. and Mcgregor, J. (1962) Occupation time laws for birth and death processes. Proc. 4th Berkeley Symp. Math. Statist. Prob. 2, 249–272.Google Scholar