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arXiv:0906.3133v2[math.PR]17Mar2010 The Functional Equation of the Smoothing

Summary: arXiv:0906.3133v2[math.PR]17Mar2010
The Functional Equation of the Smoothing
Gerold Alsmeyer
Universit¨at M¨unster
J. D. Biggins
University of Sheffield
Matthias Meiners
Uppsala universitet
March 18, 2010
Given a sequence T = (Ti)i1 of non-negative 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 fixed-point equation of the


Source: Alsmeyer, Gerold - Institut für Mathematische Statistik, Westfälische Wilhelms-Universität Münster


Collections: Mathematics