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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*
Affiliation:
Rensselaer Polytechnic Institute, Troy, New York

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.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1976 

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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, 109123.Google Scholar
[3] Crump, K. and Mode, C. (1968) A general age-dependent branching process, I. J. Math. Anal. Appl. 24, 494508.CrossRefGoogle Scholar
[4] Crump, K. and Mode, C. (1969) A general age-dependent branching process, II. J. Math. Anal. Appl. 25, 817.Google Scholar
[5] Durham, S. D. (1971), Limit theorems for a general critical branching process. J. Appl. Prob. 8, 116.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, 7488.CrossRefGoogle Scholar
[8] Holte, J. M. (1974) Extinction probability for a critical general branching process. Stoch. Proc. Appl. 2, 303309.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, 84103.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, 513540.Google Scholar
[12] Sevastyanov, B. A. (1968) Limit theorems for age-dependent branching processes. Theor. Prob. Appl. 13, 237259.Google Scholar
[13] Stone, C. (1965) On characteristic functions and renewal theory. Trans. Amer. Math. Soc. 120, 327342.CrossRefGoogle Scholar