Extension of Stein's lemma derived by using an integration by differentiation technique

TL;DR

This paper extends Stein's lemma to include Gaussian variables raised to powers, providing two proofs and revealing a connection between Hermite polynomials and the normal distribution.

Contribution

It introduces a novel extension of Stein's lemma for polynomial functions of Gaussian variables, with rigorous and constructive proofs.

Findings

Extension of Stein's lemma for Gaussian powers

Connection between Hermite polynomials and Gaussian distribution

Two different proofs provided

Abstract

We extend Stein's lemma for averages that explicitly contain the Gaussian random variable at a power. We present two proofs for this extension of Stein's lemma, with the first being a rigorous proof by mathematical induction. The alternative, second proof is a constructive formal derivation in which we express the average not as an integral, but as the action of a pseudodifferential operator defined via the Gaussian moment-generating function. In extended Stein's lemma, the absolute values of the coefficients of the probabilist's Hermite polynomials appear, revealing yet another link between Hermite polynomials and normal distribution.