 
Summary: arXiv:0906.3133v2[math.PR]17Mar2010
The Functional Equation of the Smoothing
Transform
Gerold Alsmeyer
Universit¨at M¨unster
J. D. Biggins
University of Sheffield
Matthias Meiners
Uppsala universitet
March 18, 2010
Abstract
Given a sequence T = (Ti)i1 of nonnegative random variables, a
function f on the positive halfline can be transformed to E i1 f(tTi).
We study the fixed points of this transform within the class of decreasing
functions. By exploiting the intimate relationship with general branch
ing processes, a full description of the set of solutions is established
without the moment conditions that figure in earlier studies. Since
the class of functions under consideration contains all Laplace trans
forms of probability distributions on [0, ), the results provide the full
description of the set of solutions to the fixedpoint equation of the
