Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T22:57:28.659Z Has data issue: false hasContentIssue false
  • English
  • Français

On a functional equation for general branching processes

Published online by Cambridge University Press:  14 July 2016

R. A. Doney
R. A. Doney*
Affiliation:
University of Manchester
Get access Rights & Permissions [Opens in a new window]

Abstract

If Z(t) denotes the population size in a Bellman-Harris age-dependent branching process such that a non-denenerate random variable W, then it is known that E(W) = 1 and that ϕ (u) = E(e–uW) satisfies a well-known integral equation. In this situation Athreya [1] has recently found a NASC for E(W |log W| y) <∞, for γ > 0. This paper generalizes Athreya's results in two directions. Firstly a more general class of branching processes is considered; secondly conditions are found for E(W 1 + βL(W)) < ∞ for 0 β < 1, where L is one of a class of functions of slow variation.

Keywords

BRANCHING PROCESSFUNCTIONAL EQUATION
Type
Short Communications
Information
Journal of Applied Probability , Volume 10 , Issue 1 , March 1973 , pp. 198 - 205
Copyright
Copyright © Applied Probability Trust 1973 

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. (1971) A note on a functional equation arising in Galton-Watson branching processes. J. Appl. Prob. 8, 589–598.Google Scholar
[2] Crump, K. and Mode, C. J. (1968) A general age-dependent branching process, I. J. Math . Anal. Appl. 24, 494–508.Google Scholar
[3] Crump, K. and Mode, C. J. (1969) A general age-dependent branching process, II. J . Math. Anal. Appl. 25, 8–17.Google Scholar
[4] Doney, R. A. (1972) A limit theorem for a class of supercritical branching processes. J. Appl. Prob. 9, 707–724.CrossRefGoogle Scholar
[5] Mode, C. J. (1971) Multitype branching processes. American Elsevier, New York.Google Scholar