Papers › On algebraic Stein operators for Gaussian polynomials
On algebraic Stein operators for Gaussian polynomials
Ehsan Azmoodeh, Dario Gasbarra, Robert E. Gaunt
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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=(X₁,…, X_d) has i.i.d$.$ standard Gaussian components and h∈𝕂[X] is a polynomial with coefficients in the ring 𝕂. 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 H₂₀(X₁), where Hₚ is the p-th Hermite polynomial. Some examples of Stein operators for Hₚ(X₁), p=3,4,5,6, are gathered in the Appendix and many other examples are given in the Supplementary Information.
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