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On the limiting distribution of a supercritical branching process in a random environment

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Ben Hambly
Ben Hambly*
Affiliation:
University of California, San Diego
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Abstract

We consider an increasing supercritical branching process in a random environment and obtain bounds on the Laplace transform and distribution function of the limiting random variable. There are two possibilities that can be distinguished depending on the nature of the component distributions of the environment. If the minimum family size of each is 1, the growth will be as a power depending on a parameter α. If the minimum family sizes of some are greater than 1, it will be exponential, depending on a parameter γ. We obtain bounds on the distribution function analogous to those found for the simple Galton-Watson case. It is not possible to obtain exact estimates and we are only able to obtain bounds to within ε of the parameters.

Keywords

FUNCTIONAL EQUATIONRANDOM FRACTAL

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 3 , September 1992 , pp. 499 - 518
Copyright
Copyright © Applied Probability Trust 1992 

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References

[1] Athreya, K. D. and Karlin, S. (1971) On branching processes in random environments. I. Ann. Math. Statist. 42, 1499–1520.CrossRefGoogle Scholar
[2] Athreya, K. B. and Karlin, S. (1971) On branching processes in random environments. II. Ann. Math. Statist. 42, 1843–1858.CrossRefGoogle Scholar
[3] Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
[4] Barlow, M. T. and Bass, R. F. (1989) The construction of Brownian motion on the Sierpinski carpet. Ann. Inst. H. Poincaré 25, 225–257.Google Scholar
[5] Barlow, M. T. and Perkins, E. A. (1988) Brownian motion on the Sierpinski gasket. Prob. Theory Rel. Fields 79, 543–624.Google Scholar
[6] Bingham, N. H. (1988) On the limit of a supercritical branching process. J. Appl. Prob. 25A, 215–228.CrossRefGoogle Scholar
[7] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987) Regular Variation. Cambridge University Press.Google Scholar
[8] Dubuc, S. (1971) La densité de la loi-limite d'un processus en cascade expansif. Z. Wahrscheinlichkeitsth. 19, 281–290.Google Scholar
[9] Dubuc, S. (1971) Problèmes relatifs à l'itération des fonctions suggerés par les processus en cascade. Ann. Inst. Fourier 21, 171–251.Google Scholar
[10] Durrett, R. (1988) Lecture notes on Particle Systems and Percolation. Wadsworth, Belmont, CA.Google Scholar
[11] Goettge, R. T. (1976) Limit theorems for the supercritical Galton-Watson process in varying environments. Math. Biosci. 28, 171–190.CrossRefGoogle Scholar
[12] Harris, T. E. (1948) Branching processes. Ann. Math. Statist. 19, 474–494.Google Scholar
[13] Jagers, P. (1974) Galton-Watson processes in varying environments. J. Appl. Prob. 11, 174–178.CrossRefGoogle Scholar
[14] Tanny, D. (1977) Limit theorems for branching processes in random environments. Ann. Prob. 5, 100–116.CrossRefGoogle Scholar
[15] Tanny, D. (1988) A necessary and sufficient condition for a branching process to grow like the product of its means. Stoch. Proc. Appl. 28, 123–139.CrossRefGoogle Scholar