Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T16:29:59.149Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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