Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-22T18:58:16.406Z Has data issue: false hasContentIssue false
  • English
  • Français

An age dependent branching process with variable lifetime distribution: The generation size

Published online by Cambridge University Press:  01 July 2016

Robert Fildes
Robert Fildes*
Affiliation:
Manchester Business School
Get access Rights & Permissions [Opens in a new window]

Abstract

In a branching process with variable lifetime, introduced by Fildes (1972) define Yjk(t) as the number of particles alive in generation k at time t when the initial particle is born in generation j. A limit law similar to that derived in the Bellman-Harris process is proved where it is shown that Yjk(t) suitably normalised converges in mean square to a random variable which is the limit random variable of Znm–n in the Galton-Watson process (m is the mean number of particles born).

Keywords

AGE-DEPENDENTBRANCHING PROCESSMULTI-TYPEVARIABLE LIFETIMEGENERATIONBELLMAN-HARRIS
Type
Research Article
Information
Advances in Applied Probability , Volume 6 , Issue 2 , June 1974 , pp. 291 - 308
Copyright
Copyright © Applied Probability Trust 1974 

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] Cramér, H. (1962) Random Variables and Probability Distributions. 2nd ed. Cambridge University Press.Google Scholar
[2] Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. II. Wiley, New York.Google Scholar
[3] Fildes, R. (1972) An age-dependent branching process with variable lifetime distribution. Adv. Appl. Prob. 4, 453–474.CrossRefGoogle Scholar
[4] Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
[5] Katz, M. (1963) The probability in the tail of a distribution. Ann. Math. Statist. 34, 312–318.Google Scholar
[6] Kharlamov, B. P. (1968) On properties of branching processes with an arbitrary set of particle types. Theor. Probability Appl. 13, 84–98.Google Scholar
[7] Kharlamov, B. P. (1969) On the generation numbers of particles in a branching process with overlapping generations. Theor. Probability Appl. 14, 44–50.Google Scholar
[8] Kharlamov, B. P. (1969) The number of generations in a branching process with an arbitrary set of particle types. Theor. Probability Appl. 14, 432–449.CrossRefGoogle Scholar
[9] Loève, M. (1963) Probability Theory. 3rd ed. Van Nostrand, Princeton, New Jersey.Google Scholar
[10] Martin-Löf, A. (1966) A limit theorem for the size of the nth generation of an age-dependent branching process. J. Math. Anal. Appl. 15, 273–279.Google Scholar