An asymptotic approach to proving sufficiency of Stein characterisations

TL;DR

This paper introduces an asymptotic method to prove the sufficiency of Stein characterisations for various distributions, expanding the applicability of Stein's method in probability theory.

Contribution

The paper develops a general asymptotic approach to establish the sufficiency of Stein characterisations for linear differential operators with polynomial coefficients, including new results for Hermite and product distributions.

Findings

Proves all linear coefficient Stein operators characterise their distributions.

Verifies polynomial Stein operators of degree at most two are characterising.

Establishes characterisations for Hermite and product distributions with minimal degree Stein operators.

Abstract

In extending Stein's method to new target distributions, the first step is to find a Stein operator that suitably characterises the target distribution. In this paper, we introduce a widely applicable technique for proving sufficiency of these Stein characterisations, which can be applied when the Stein operators are linear differential operators with polynomial coefficients. The approach involves performing an asymptotic analysis to prove that only one characteristic function satisfies a certain differential equation associated to the Stein characterisation. We use this approach to prove that all Stein operators with linear coefficients characterise their target distribution, and verify on a case-by-case basis that all polynomial Stein operators in the literature with coefficients of degree at most two are characterising. For $X$ denoting a standard Gaussian random variable and $H_p$ the $p$-th Hermite polynomial, we also prove, amongst other examples, that the Stein operators for $H_p(X)$, $p=3,4,\ldots,8$, with coefficients of minimal possible degree characterise their target distribution, and that the Stein operators for the products of $p=3,4,\ldots,8$ independent standard Gaussian random variables are characterising (in both settings the Stein operators for the cases $p=1,2$ are already known to be characterising). We leverage our Stein characterisations of $H_3(X)$ and $H_4(X)$ to derive characterisations of these target distributions in terms of iterated Gamma operators from Malliavin calculus, that are natural in the context of the Malliavin-Stein method.