The functional equation of the smoothing transform

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.