A simple method to find all solutions to the functional equation of the smoothing transform

Gerold Alsmeyer (1) , Bastien Mallein (2)

1 Institut für Mathematische Statistik [Munster]
2 LAGA - Laboratoire Analyse, Géométrie et Applications

Given a nonincreasing null sequence $T = (T_j, j \geqslant 1)$ of nonnegative random variables, the well-known functional equation $$ f(t)=\textstyle\mathbb{E}\left(\prod_{j\geqslant 1}f(tT_{j})\right), $$ related to the so-called smoothing transform and a min-type variant, is reconsidered within the class of nonnegative and nonincreasing functions. In order to characterize all solutions within this class, we provide a new three-step method which does not only considerably simplify earlier approaches but also works under weaker, close to optimal conditions. Furthermore, we expect it to work as well in more general setups like random environment. At the end of the article, we also give a one-to-one correspondence between those solutions that are Laplace transforms and thus correspond to the fixed points of the smoothing transform and certain fractal random measures. The latter are defined on the boundary of a weighted tree related to an associated branching random walk.

Domaines: Probabilités [math.PR]

Gerold Alsmeyer, Bastien Mallein. A simple method to find all solutions to the functional equation of the smoothing transform. 2019. ⟨hal-02177977v1⟩