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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 Z0 = 1 and let G denote the distribution function of Z1. 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
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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References

[1] Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.Google Scholar
[2] Bingham, N. H. and Teugels, J. L. (1975) Duality for regularly varying functions. Quart. J. Math. Oxford (3) 26, 333–353.CrossRefGoogle Scholar
[3] Cohn, H. (1977) Almost sure convergence of branching processes. Z. Wahrscheinlichkeitsth. 38, 73–81.CrossRefGoogle Scholar
[4] Darling, D. H. (1970) The Galton–Watson process with infinite mean. J. Appl. Prob. 7, 455–456.Google Scholar
[5] Davies, P. L. (1974) Eine Klasse nirgends differenzierbarer stochastischer Prozesse mit stationären Gaußschen Zuwachsen. Math. Nachr. 63, 197–204.Google Scholar
[6] Davies, P. L. (1976) Tail probabilities for positive random variables with entire characteristic functions of very regular growth. Z. angew. Math. Mech. 56, T334–T336.Google Scholar
[7] Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
[8] Grey, D. R. (1977) Almost sure convergence in Markov branching processes with infinite mean. J. Appl. Prob. 14, 702–716.Google Scholar
[9] Hudson, I. L. and Seneta, E. (1977) A note on simple branching processes with infinite mean. J. Appl. Prob. 14, 836–842.Google Scholar
[10] Kallenberg, O. (1976) Some applications of Feller's dominated variation. Tagung ‘Mathematische Stochastik’, Oberwolfach.Google Scholar
[11] Schuh, H.-J. and Barbour, A. D. (1977) On the asymptotic behaviour of branching processes with infinite mean. Adv. Appl. Prob. 9, 681–723.Google Scholar
[12] Seneta, E. (1969) Functional equations and the Galton–Watson process. Adv. Appl. Prob. 1, 1–42.Google Scholar
[13] Seneta, E. (1973) The simple branching process with infinite mean, I. J. Appl. Prob. 10, 206–212.CrossRefGoogle Scholar
[14] Seneta, E. (1976) Regularly Varying Functions. Lecture Notes in Mathematics 508, Springer-Verlag, Berlin.Google Scholar