Explicit fixed points of the smoothing transformation

Abstract

We deal with the equation $Y\stackrel{\mathrm{d}}{=}\frac{1}{b}{\sum }_{1\le j\le N}{W}_j{Y}_j$, 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 $Y$.