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Asymptotic properties of super-critical branching processes II: Crump-Mode and Jirina processes

Published online by Cambridge University Press:  01 July 2016

N. H. Bingham  and
R. A. Doney
N. H. Bingham
Affiliation:
Westfield College, London
R. A. Doney
Affiliation:
University of Manchester
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Abstract

We obtain results connecting the distribution of the random variables Y and W in the supercritical generalized branching processes introduced by Crump and Mode. For example, if β > 1, EYβ and EWβ converge or diverge together and regular variation of the tail of one of Y, W with non-integer exponent β > 1 is equivalent to regular variation of the other. We also prove analogous results for the continuous-time continuous state-space branching processes introduced by Jirina.

Keywords

GENERAL BRANCHING PROCESSSUPERCRITICALMOMENTSTAIL BEHAVIOURREGULAR VARIATIONLAPLACE-STIELTJES TRANSFORM
Type
Research Article
Information
Advances in Applied Probability , Volume 7 , Issue 1 , March 1975 , pp. 66 - 82
Copyright
Copyright © Applied Probability Trust 1975 

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References

[1] Athreya, K. B. and Ney, P. E. (1973) Branching Processes. Springer, Berlin.Google Scholar
[2] Bingham, N. H. and Doney, R. A. (1974) Asymptotic properties of supercritical branching processes, I: the Galton-Watson process. J. Appl. Prob. 6, 711–731.Google Scholar
[3] Crump, K. and Mode, C. J. (1968) A general age-dependent branching process I. J. Math. Anal. Appl. 24, 494–508.Google Scholar
[4] Crump, K. and Mode, C. J. (1969) A general age-dependent branching process II. J. Math. Anal. Appl. 25, 8–17.Google Scholar
[5] Doney, R. A. (1972) A limit theorem for a class of supercritical branching processes. J. Appl. Prob. 9, 707–724.Google Scholar
[6] Doney, R. A. (1973) On a functional equation for general branching processes. J. Appl. Prob. 10, 198–205.CrossRefGoogle Scholar
[7] Ganuza, E. (1975) On supercritical branching processes. J. Appl. Prob. To appear.Google Scholar
[8] Grey, D. R. (1975) Asymptotic properties of continuous-time, continuous state-space branching processes. To appear.Google Scholar
[9] Jagers, P. (1969) A general stochastic model for population development. Skand. Aktuarietidskr. 1–2, 84–103.Google Scholar
[10] Jirina, M. (1958) Stochastic branching processes with continuous state space. Czech. Math. J. 8, 292–313.Google Scholar