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Sur Une Équation Fonctionnelle Et SES Applications: Une Extension Du Théorème De Kesten-Stigum Concernant Des Processus De Branchement

Part of: Markov processes Special processes

Published online by Cambridge University Press:  01 July 2016

Quansheng Liu
Quansheng Liu*
Affiliation:
Universit de Rennes 1
*
∗ Postal address: Institut Mathematique de Rennes, Universite de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France. Email: [email protected]
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Abstract

Given a random integer N ≧ 0 and a sequence of random variables Ai ≧ 0, we define a transformation T on the class of probability measures on [0, ∞) by letting Tμ be the distribution of , where {Zi} are independent random variables with distribution μ, which are independent of N and of {Ai} as well. We obtain the optimal conditions for the functional equation μ = Tμ to have a non-trivial solution of finite mean, and we study the existence of moments of the solutions. The work unifies the corresponding theorems of Kesten-Stigum concerning the Galton-Watson process, of Biggins for branching random walks, of Kahane-Peyrière for a model of turbulence of Yaglom made precise by Mandelbrot, and of Durrett-Liggett for the study of invariant measures for certain infinite particle systems.

Keywords

BRANCHING PROCESSESBRANCHING RANDOM WALKSFUNCTIONAL EQUATIONMANDELBROT'S MARTINGALE

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 2 , June 1997 , pp. 353 - 373
Copyright
Copyright © Applied Probability Trust 1997 

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