Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-29T07:15:33.100Z Has data issue: false hasContentIssue false
  • English
  • Français

On the extinction time distribution of a branching process in varying environments

Published online by Cambridge University Press:  01 July 2016

Tetsuo Fujimagari
Tetsuo Fujimagari*
Affiliation:
Kanazawa University
*
∗Postal address: Kanazawa University, College of Liberal Arts, 1-1 Marunouchi, Kanazawa 920, Japan.
Get access Rights & Permissions [Opens in a new window]

Abstract

The extinction time distributions of a class of branching processes in varying environments are considered. We obtain (i) sufficient conditions for the extinction probability q = 1 or q < 1; (ii) asymptotic formulae for the tail probability of the extinction time if q = 1; and (iii) upper bounds for 1 – q if q < 1. To derive these results, we give upper and lower bounds for the tail probability of the extinction time. For the proofs, we use a method that compares probability generating functions with fractional linear generating functions.

Keywords

BRANCHING PROCESSVARYING ENVIRONMENTSEXTINCTION TIME DISTRIBUTIONEXTINCTION PROBABILITYFRACTIONAL LINEAR FUNCTIONFRACTIONAL LINEAR GENERATING FUNCTION
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 2 , June 1980 , pp. 350 - 366
Copyright
Copyright © Applied Probability Trust 1980 

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

Agresti, A. (1974) Bounds on the extinction time distribution of a branching process. Adv. Appl. Prob. 6, 322–335.Google Scholar
Agresti, A. (1975) On the extinction times of varying and random environment branching processes. J. Appl. Prob. 12, 39–46.Google Scholar
Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer, Berlin.Google Scholar
Church, J. D. (1971) On infinite composition products of probability generating functions. Z. Wahrscheinlichkeitsth. 19, 243–256.Google Scholar
Jagers, P. (1975) Branching Processes with Biological Applications. Wiley, New York.Google Scholar
Jirina, M. (1976) Extinction of non-homogeneous Galton–Watson processes. J. Appl. Prob. 13, 132–137.Google Scholar