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

Given a random integerN≧ 0 and a sequence of random variablesAi≧ 0, we define a transformation T on the class of probability measures on [0, ∞) by lettingTμ 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.