A unified approach to Stein characterizations

TL;DR

This paper introduces a unified framework for Stein characterizations of probability distributions, generalizing classical methods and enabling the derivation of new characterizations through a parametric interpretation.

Contribution

It provides a general methodology that unifies existing Stein characterizations, clarifies minimal conditions, and facilitates the creation of new characterizations.

Findings

Unified framework for Stein characterizations

Clear identification of minimal conditions

Method for constructing new Stein characterizations

Abstract

This article deals with Stein characterizations of probability distributions. We provide a general framework for interpreting these in terms of the parameters of the underlying distribution. In order to do so we introduce two concepts (a class of functions and an operator) which generalize those which were developed in the 70's by Charles Stein and Louis Chen for characterizing the Gaussian and the Poisson distributions. Our methodology (i) allows for writing many (if not all) known univariate Stein characterizations, (ii) permits to identify clearly minimal conditions under which these results hold and (iii) provides a straightforward tool for constructing new Stein characterizations. Our parametric interpretation of Stein characterizations also raises a number of questions which we outline at the end of the paper.