On algebraic Stein operators for Gaussian polynomials

Ehsan Azmoodeh, Dario Gasbarra, Robert E. Gaunt

Published 2019-12-10Version 1

The first essential ingredient to build up Stein's method for a continuous target distribution is to identify a so-called \textit{Stein operator}, namely a linear differential operator with polynomial coefficients. In this paper, we introduce the notion of \textit{algebraic} Stein operators (see Definition \ref{def:algebraic-Stein-Operator}), and provide a novel algebraic method to find \emph{all} the algebraic Stein operators up to a given order and polynomial degree for a target random variable of the form $Y=h(X)$, where $X=(X_1,\dots, X_d)$ has i.i.d$.$ standard Gaussian components and $h\in \KK[X]$ is a polynomial with coefficients in the ring $\KK$. Our approach links the existence of an algebraic Stein operator with \textit{null controllability} of a certain linear discrete system. A \texttt{MATLAB} code checks the null controllability up to a given finite time $T$ (the order of the differential operator), and provides all \textit{null control} sequences (polynomial coefficients of the differential operator) up to a given maximum degree $m$. This is the first paper that connects Stein's method with computational algebra to find Stein operators for highly complex probability distributions, such as $H_{20}(X_1)$, where $H_p$ is the $p$-th Hermite polynomial. A number of examples of Stein operators for $H_p(X_1)$, $p=3,4,5,6,7,8,10,12$, are gathered in the extended Appendix of this arXiv version. We also introduce a widely applicable approach to proving that Stein operators characterise the target distribution, and use it to prove, amongst other examples, that the Stein operators for $H_p(X_1)$, $p=3,\ldots,8$, with minimum possible maximal polynomial degree $m$ characterise their target distribution.