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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
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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 

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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