On perpetuities related to the size-biased distributions

TL;DR

This paper investigates a special class of perpetuities linked to size-biased distributions, establishing conditions for their existence and uniqueness through connections to Poisson shot noise transforms.

Contribution

It provides necessary and sufficient conditions for the existence and uniqueness of these perpetuities, highlighting their relation to fixed points of Poisson shot noise transforms.

Findings

Conditions for existence and uniqueness of perpetuities

Connection between perpetuities and Poisson shot noise transforms

Characterization of fixed points in the context of size-biased distributions

Abstract

We study perpetuities of a special type related to the size-biased distributions. Necessary and sufficient conditions of their existence and uniqueness are obtained. A crucial point in proving all results is a close connection between perpetuities treated in the paper and fixed points of so-called Poisson shot noise transforms.