Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-04T21:37:22.467Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

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. (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