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September 2012 The functional equation of the smoothing transform
Gerold Alsmeyer, J. D. Biggins, Matthias Meiners
Ann. Probab. 40(5): 2069-2105 (September 2012). DOI: 10.1214/11-AOP670

Abstract

Given a sequence $T=(T_{i})_{i\geq1}$ of nonnegative random variables, a function $f$ on the positive halfline can be transformed to $\mathbb{E}\prod_{i\geq1}f(tT_{i})$. 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 $[0,\infty)$, the results provide the full description of the set of solutions to the fixed-point equation of the smoothing transform, $X\stackrel{d}{=}\sum_{i\geq1}T_{i}X_{i}$, where $\stackrel{d}{=}$ denotes equality of the corresponding laws, and $X_{1},X_{2},\ldots$ is a sequence of i.i.d. copies of $X$ independent of $T$. Further, since left-continuous survival functions are covered as well, the results also apply to the fixed-point equation $X\stackrel{d}{=}\inf\{X_{i}/T_{i} : i\geq1,T_{i}>0\}$. Moreover, we investigate the phenomenon of endogeny in the context of the smoothing transform and, thereby, solve an open problem posed by Aldous and Bandyopadhyay.

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Gerold Alsmeyer. J. D. Biggins. Matthias Meiners. "The functional equation of the smoothing transform." Ann. Probab. 40 (5) 2069 - 2105, September 2012. https://doi.org/10.1214/11-AOP670

Information

Published: September 2012
First available in Project Euclid: 8 October 2012

zbMATH: 1266.39022
MathSciNet: MR3025711
Digital Object Identifier: 10.1214/11-AOP670

Subjects:
Primary: 39B22
Secondary: 60E05 , 60G42 , 60J85

Keywords: branching process , Branching random walk , Choquet–Deny-type functional equation , endogeny , fixed point , general branching process , Multiplicative martingales , Smoothing transformation , stochastic fixed-point equation , Weibull distribution , weighted branching

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.40 • No. 5 • September 2012
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