Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T12:14:45.483Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic probabilities in a critical age-dependent branching process

Published online by Cambridge University Press:  14 July 2016

Howard J. Weiner
Howard J. Weiner*
Affiliation:
University of California at Davis
Get access Rights & Permissions [Opens in a new window]

Abstract

Let Z(t) denote the number of cells alive at time t in a critical Bellman-Harris age-dependent branching process, that is, where the mean number of offspring per parent is one. A comparison method is used to show for k ≧ 1, and a high-order moment condition on G(t), where G(t) is the cell lifetime distribution, that lim t→∞t2P[Z(t) = k] = ak > 0, where {ak} are constants.

The method is also applied to the total progeny in the critical process.

Keywords

CRITICAL BRANCHING PROCESSASYMPTOTIC PROBABILITYAGE-DEPENDENT
Type
Research Papers
Information
Journal of Applied Probability , Volume 13 , Issue 3 , September 1976 , pp. 476 - 485
Copyright
Copyright © Applied Probability Trust 1976 

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.)

References

[1] Athreya, K. B. and Karlin, S. (1967) Limit theorems for the split times of branching processes. J. Math. Mech. 17, 257–278.Google Scholar
[2] Athreya, K. and Ney, P. (1972) Branching Processes. Springer-Verlag, New York.Google Scholar
[3] Chover, J. and Ney, P. (1968) The non-linear renewal equation. J. Anal. Math. 21, 381–413.Google Scholar
[4] Weiner, H. (1966) On age-dependent branching processes. J. Appl. Prob. 3, 383–402.Google Scholar