{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/on-algebraic-stein-operators-for-gaussian","title":"On algebraic Stein operators for Gaussian polynomials","arxiv_id":"1912.04605","date":"2019-12-10","proceeding":null,"authors":["Ehsan Azmoodeh","Dario Gasbarra","Robert E. Gaunt"],"abstract":"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 $h\\in \\mathbb{K}[X]$ is a polynomial with coefficients in the ring $\\mathbb{K}$. 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 $H_{20}(X_1)$, where $H_p$ is the $p$-th Hermite polynomial. Some examples of Stein operators for $H_p(X_1)$, $p=3,4,5,6$, are gathered in the Appendix and many other examples are given in the Supplementary Information.","url_abs":"https://arxiv.org/abs/1912.04605v4","url_pdf":"https://arxiv.org/pdf/1912.04605v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"on-algebraic-stein-operators-for-gaussian","repo_url":"https://github.com/gasbarra/algebraic-stein-equations","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}