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Another look at the finite mean supercritical Bienaymé–Galton–Watson process

Published online by Cambridge University Press:  14 July 2016

Harry Cohn
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Abstract

Let {Zn} be a finite mean supercritical Bienaymé– Galton–Watson process. It is known that there exist norming constants {Cn} such that {Zn/Cn} converges almost surely to a limit W. Also there is a whole literature concerning properties of {Cn} and W. We attempt a new approach to the limit theory of {Zn} by relating it to the theory of sums of independent and identically distributed random variables.

Keywords

BIENAYMÉ–GALTON–WATSON PROCESSOFFSPRING DISTRIBUTIONSUPERCRITICALITYSUMS OF INDEPENDENT RANDOM VARIABLESQUANTILES
Type
Part 6 — Stochastic Processes
Information
Journal of Applied Probability , Volume 19 , Issue A , 1982 , pp. 307 - 312
Copyright
Copyright © 1982 Applied Probability Trust 

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References

[1] Athreya, K. B. (1971) A note on a functional equation arising in Galton-Watson processes. J. Appl. Prob. 8, 589–598.Google Scholar
[2] Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, New York.Google Scholar
[3] Cohn, H. (1977) Almost sure convergence of branching processes. Z. Wahrscheinlichkeitsth. 38, 73–81.Google Scholar
[4] Cohn, H. (1980) Norming constants for the finite mean supercritical Bellman-Harris process. University of Melbourne Department of Statistics Research Report Series No. 6.Google Scholar
[5] Dubuc, S. (1971) Problèmes relatifs à l'itération de fonctions suggérés par les processus en cascade. Ann. Inst. Fourier (Grenoble) 21, 171–251.Google Scholar
[6] Grey, D. R. (1980) A new look at convergence of branching processes. Ann. Prob. 8, 377–381.CrossRefGoogle Scholar
[7] Feller, W. (1971) An Introduction to Probability Theory and its Applications , Vol. II. Wiley, New York.Google Scholar
[8] Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
[9] Heyde, C. C. (1970) Extension of a result by Seneta for the supercritical branching process. Ann. Math. Statist. 41, 739–742.Google Scholar
[10] Kesten, H. and Stigum, B. P. (1966) A limit theorem for multi-dimensional Galton-Watson processes. Ann. Math. Statist. 37, 1211–1223.Google Scholar
[11] Seneta, E. (1968) On recent theorems concerning the supercritical Galton–Watson process. Ann. Math. Statist. 39, 2098–2102.Google Scholar
[12] Seneta, E. (1974) Characterizations by functional equations of branching processes limit laws. In Statistical Distribution in Scientific Work , 3, ed. Patil, G. et al., 249–254, Calgary.Google Scholar
[13] Seneta, ?. (1974) Regularly varying functions in the theory of simple branching processes. Adv. Appl. Prob. 6, 408–420.CrossRefGoogle Scholar