Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-12-01T00:15:57.081Z Has data issue: false hasContentIssue false
  • English
  • Français

A generalization of Goldstein's comparison lemma and the exponential limit law in critical Crump-Mode-Jagers branching processes

Published online by Cambridge University Press:  01 July 2016

John M. Holte
John M. Holte*
Affiliation:
Rensselaer Polytechnic Institute, Troy, New York
Get access Rights & Permissions [Opens in a new window]

Abstract

Let Z(t) be the population size at time t in a general age-dependent branching process (as defined by Crump and Mode, or Jagers) in which the number N of offspring of a parent has expected value 1 (critical case). Assuming positivity and finiteness of the second moments of N, of the lifespan distribution and of the expected number of births per parent as a function of age (also assumed to be strongly non-lattice), the distribution of Z(t)/t conditioned on non-extinction at time t is asymptotically exponential. The main step in the proof is a comparison lemma for the probability generating functions of Z(t) and of the embedded generation process.

Keywords

BRANCHING PROCESSESCRITICALCONDITIONED LIMIT LAWEMBEDDED GALTON–WATSON PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 8 , Issue 1 , March 1976 , pp. 88 - 104
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 Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.Google Scholar
[2] Baum, L. F. and Katz, M. (1965) Convergence rates in the law of large numbers. Trans. Amer. Math. Soc. 120, 109–123.Google Scholar
[3] Crump, K. and Mode, C. (1968) A general age-dependent branching process, I. J. Math. Anal. Appl. 24, 494–508.CrossRefGoogle Scholar
[4] Crump, K. and Mode, C. (1969) A general age-dependent branching process, II. J. Math. Anal. Appl. 25, 8–17.Google Scholar
[5] Durham, S. D. (1971), Limit theorems for a general critical branching process. J. Appl. Prob. 8, 1–16.Google Scholar
[6] Feller, W. (1966) An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley, New York.Google Scholar
[7] Goldstein, M. I. (1971) Critical age-dependent branching processes: single and multitype. Z. Wahrscheinlichkeitsth. 17, 74–88.CrossRefGoogle Scholar
[8] Holte, J. M. (1974) Extinction probability for a critical general branching process. Stoch. Proc. Appl. 2, 303–309.Google Scholar
[9] Holte, J. M. (1974) Limit theorems for critical general branching processes. , University of Wisconsin, Madison.Google Scholar
[10] Jagers, P. (1969) A general stochastic model for population development. Skand. Aktuarietidskr. 52, 84–103.Google Scholar
[11] Kesten, J., Ney, P. and Spitzer, F. (1966) The Galton–Watson process with mean one and finite variance. Theor. Prob. Appl. 11, 513–540.Google Scholar
[12] Sevastyanov, B. A. (1968) Limit theorems for age-dependent branching processes. Theor. Prob. Appl. 13, 237–259.Google Scholar
[13] Stone, C. (1965) On characteristic functions and renewal theory. Trans. Amer. Math. Soc. 120, 327–342.CrossRefGoogle Scholar