Fixed points of inhomogeneous smoothing transforms

TL;DR

This paper characterizes the existence and structure of fixed points for inhomogeneous smoothing transforms, providing necessary and sufficient conditions and linking solutions to the homogeneous case.

Contribution

It offers a complete characterization of fixed points for inhomogeneous smoothing transforms, including necessary and sufficient conditions and a correspondence with the homogeneous case.

Findings

Established a necessary and sufficient condition for fixed point existence.

Provided an explicit correspondence with solutions of the homogeneous equation.

Achieved a full characterization of fixed points under mild assumptions.

Abstract

We consider the inhomogeneous version of the fixed-point equation of the smoothing transformation, that is, the equation $X \stackrel{d}{=} C + \sum_{i \geq 1} T_i X_i$, where $\stackrel{d}{=}$ means equality in distribution, $(C,T_1,T_2,...)$ is a given sequence of non-negative random variables and $X_1,X_2,...$ is a sequence of i.i.d.\ copies of the non-negative random variable $X$ independent of $(C,T_1,T_2,...)$. In this situation, $X$ (or, more precisely, the distribution of $X$) is said to be a fixed point of the (inhomogeneous) smoothing transform. In the present paper, we give a necessary and sufficient condition for the existence of a fixed point. Further, we establish an explicit one-to-one correspondence with the solutions to the corresponding homogeneous equation with C=0. Using this correspondence, we present a full characterization of the set of fixed points under mild assumptions.