Explicit fixed points of the smoothing transformation

TL;DR

This paper investigates the fixed points of a smoothing transformation equation, providing explicit formulas for the distribution of solutions in certain cases, which are typically obtained as martingale limits.

Contribution

It offers explicit solutions for the distribution of fixed points of the smoothing transformation, expanding beyond the usual martingale limit approach.

Findings

Explicit formulas for the law of Y in specific cases

Conditions under which explicit solutions can be derived

Extension of known fixed point solutions in smoothing transformations

Abstract

We deal with the equation $Y \stackrel{\rm d}{=} \frac{1}{b} \sum_{1\le j\le N} W_jY_j$, where the unknown is the distribution of $Y$, the variables in the right hand side are independent, the $Y_j$ are equidistributed with $Y$, $N$ is an integer valued random variable, and the $W_j$ are equidistributed, nonnegative and of expectation~1. Usually a solution is obtained as the limit of a martingale. In some cases we give an explicit formula for the law of $Y$.