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On the limit of a supercritical branching process

Published online by Cambridge University Press:  14 July 2016

N. H. Bingham
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Abstract

Let W be the usual almost-sure limit random variable in a supercritical simple branching process; we study its tail behaviour. For the left tail, we distinguish two cases, the ‘Schröder' and ‘Böttcher' cases; both appear in work of Harris and Dubuc. The Schröder case is related to work of Karlin and McGregor on embeddability in continuous-time (Markov) branching processes. New results are obtained for the Böttcher case; there are links with recent work of Barlow and Perkins on Brownian motion on a fractal. The right tail is also considered. Use is made of recent progress in Tauberian theory.

Keywords

FUNCTIONAL EQUATIONSITERATIONTAUBERIAN THEORYEMBEDDABILITYBROWNIAN MOTIONFRACTALINFINITE DIVISIBILITY
Type
Part 6 - The Analysis of Stochastic Phenomena
Information
Journal of Applied Probability , Volume 25 , Issue A , 1988 , pp. 215 - 228
Copyright
Copyright © Applied Probability Trust 1988 

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References

Asmussen, S. (1978) Some martingale methods in the limit theory of supercritical branching processes. Advances in Probability 5, ed. Joffe, A. and Ney, P. E., Marcel Dekker, New York, 1–26.Google Scholar
Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, New York.Google Scholar
Barlow, M. T. and Perkins, E. (1987) Brownian motion on the Sierpinski gasket. To appear.Google Scholar
Biggins, J. D. and Shanbhag, D. N. (1981) Some divisibility problems in branching processes. Math. Proc. Camb. Phil. Soc. 90, 321–330.CrossRefGoogle Scholar
Bingham, N. H. (1976) Continuous branching processes and spectral positivity. Stoch. Proc. Appl. 4, 217–242.CrossRefGoogle Scholar
Bingham, N. H. and Doney, R. A. (1974) Asymptotic properties of supercritical branching processes. I. The Galton-Watson process. Adv. Appl. Prob. 6, 711–731.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987) Regular Variation. Encyclopaedia of Mathematics and its Applications 27, Cambridge University Press.Google Scholar
De Bruijn, N. G. (1959) Pairs of slowly oscillating functions occurring in asymptotic problems concerning the Laplace transform. Nieuw Arch. Wiskunde (3) 7, 20–26.Google Scholar
Dubuc, S. (1971a) Problèmes relatifs à l'itération des fonctions suggérées par les processus en cascade. Ann. Inst. Fourier 21, 171–251.Google Scholar
Dubuc, S. (1971b) La densité de la loi-limite d'un processus en cascade expansif. Z. Wahrschleinlichkeitsth. 19, 281–290.Google Scholar
Dubuc, S. (1982) Étude théorique et numérique de la fonction de Karlin–McGregor. J. Analyse Math. 42, 15–37.Google Scholar
Falconer, K. J. (1986) The Geometry of Fractal Sets. Cambridge Tracts 85, Cambridge University Press.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Goldstein, S. (1987) Random walks and diffusions on fractals. To appear.Google Scholar
Grey, D. R. (1975) Two necessary conditions for embeddability of a Galton-Watson branching process. Math. Proc. Camb. Phil. Soc. 78, 339–343.Google Scholar
Grey, D. R. (1980) A new look at convergence of branching processes. Ann. Prob. 8, 377–380.Google Scholar
De Haan, L. and Stadtmüller, U. (1985) Dominated variation and related concepts and Tauberian theorems for Laplace transforms. J. Math. Anal. Appl. 108, 344–365.CrossRefGoogle Scholar
Harris, T. E. (1948) Branching processes. Ann. Math. Statist. 41, 474–494.CrossRefGoogle Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
Heyde, C. C. (1970) Extension of a result of Seneta for the supercritical Galton-Watson process. Ann. Math. Statist. 41, 739–742.Google Scholar
Hoppe, F. M. and Seneta, E. (1978) Analytic methods for discrete branching processes. Advances in Probability 5, ed. Joffe, A. and Ney, P. E., Marcel Dekker, New York, 219–262.Google Scholar
Karlin, S. and Mcgregor, J. (1968a) Embeddability of discrete time simple branching processes into continuous time branching processes. Trans. Amer. Math. Soc. 132, 115–136.Google Scholar
Karlin, S. and Mcgregor, J. (1968b) Embedding iterates of analytic functions with two fixed points into continuous groups. Trans. Amer. Math. Soc. 132, 137–145.Google Scholar
Kasahara, Y. (1978) Tauberian theorems of exponential type. J. Math. Kyoto Univ. 18, 209–219.Google Scholar
Kuczma, M. (1968) Functional Equations in a Single Variable. PWN, Warsaw.Google Scholar
Kusuoka, S. (1987) A diffusion process on a fractal. To appear.Google Scholar
Levy, P. (1937) Théorie de l'addition des variables aléatoires. Gauthier-Villars, Paris.Google Scholar
Montel, P. (1957) Leçons sur les récurrences et leurs applications. Gauthier-Villars, Paris.Google Scholar
Seneta, E. (1968) On recent theorems concerning the supercritical Galton-Watson process. Ann. Math. Statist. 39, 2098–2102.Google Scholar
Seneta, E. (1969) Functional equations and the Galton-Watson process. Adv. Appl. Prob. 1, 1–42.CrossRefGoogle Scholar
Szekeres, G. (1958) Regular iteration of real and complex functions. Acta Math. 100, 203–258.CrossRefGoogle Scholar