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 Σi=1N AiZi, 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.
