Characterization of Probability Distributions via Functional Equations of Power-Mixture Type

TL;DR

This paper investigates power-mixture functional equations related to probability distributions, providing conditions for unique solutions, extending previous results, and answering a question posed by Pitman and Yor in 2003.

Contribution

It offers new and extended conditions for the uniqueness of solutions to power-mixture functional equations, advancing the characterization of probability distributions.

Findings

Established necessary and sufficient conditions for solution uniqueness.

Extended and improved previous results on functional equations of compound-exponential and compound-Poisson types.

Provided an affirmative answer to a question by Pitman and Yor (2003).

Abstract

We study power-mixture type functional equations in terms of Laplace-Stieltjes transforms of probability distributions. These equations arise when studying distributional equations of the type Z = X + TZ, where T is a known random variable, while the variable Z is defined via X, and we want to `find' X. We provide necessary and sufficient conditions for such functional equations to have unique solutions. The uniqueness is equivalent to a characterization property of a probability distribution. We present results which are either new or extend and improve previous results about functional equations of compound-exponential and compound-Poisson types. In particular, we give another affirmative answer to a question posed by J. Pitman and M. Yor in 2003. We provide explicit illustrative examples and deal with related topics.