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The simple branching process: a note on convergence when the mean is infinite

Published online by Cambridge University Press:  14 July 2016

P. L. Davies
P. L. Davies*
Affiliation:
Universität Essen–Gesamthochschule
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Abstract

Let denote the simple branching process with Z 0 = 1 and let G denote the distribution function of Z 1 . Suppose G satisfies x −α−γ(x)≦1 − G(x) ≦ x −α+γ(x) for large x, where (i) 0 < α < 1, (ii) γ (x) is non-negative and non-increasing, (iii) xγ (x) is non-decreasing and (iv) Then limn→∞α n log (Zn + 1) converges almost surely to a non-degenerate finite random variable W satisfying P(W = 0) = q = probability of extinction of the process.

Keywords

SIMPLE BRANCHING PROCESSINFINITE MEANALMOST SURE CONVERGENCE

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 15 , Issue 3 , September 1978, pp. 466 - 480
Copyright
Copyright © Applied Probability Trust 1978 

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