The first essential ingredient to build up Stein’s method for a continuous target
distribution is to identify a so-called Stein operator, namely a linear differential operator
with polynomial coefficients. In this paper, we introduce the notion of algebraic Stein
operators , and provide a novel algebraic method to find 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 , . . . , X_d) has i.i.d. standard Gaussian components
and h in K[X] is a polynomial with coefficients in the ring K. Our approach links the
existence of an algebraic Stein operator with null controllability of a certain linear discrete
system. 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. Joint work with Ehsan Azmoodeh (Liverpool) and Rober Gaunt (Manchester).
