The Functional Equation of the Smoothing Transform by Gerold Alsmeyer,

Abstract

Given a sequence T = (Ti)i≥1 of nonnegative random variables, a function f on the positive halfline can be transformed to E ∏ i≥1 f (tTi). We study the fixed points of this transform within the class of decreasing functions. By exploiting the intimate relationship with general branching 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 transforms of probability distributions on [0,∞), the results provide the full description of the set of solutions to the

Cite this paper

@inproceedings{Biggins2012TheFE, title={The Functional Equation of the Smoothing Transform by Gerold Alsmeyer,}, author={John D. Biggins and Matthias Meiners}, year={2012} }