Stein's method, smoothing and functional approximation

TL;DR

This paper extends Stein's method for Gaussian process approximation by introducing an infinite dimensional smoothing inequality, broadening the class of functionals that can be analyzed at the expense of some precision.

Contribution

It proves a new Gaussian smoothing inequality in infinite dimensions, allowing for a wider range of functionals in Gaussian approximation with controlled bounds.

Findings

Enables approximation for Lipschitz functionals and indicator functions

Provides bounds involving smooth test functions and boundary probabilities

Expands applicability of Stein's method in high-dimensional settings

Abstract

Stein's method for Gaussian process approximation can be used to bound the differences between the expectations of smooth functionals $h$ of a c\`adl\`ag random process $X$ of interest and the expectations of the same functionals of a well understood target random process $Z$ with continuous paths. Unfortunately, the class of smooth functionals for which this is easily possible is very restricted. Here, we prove an infinite dimensional Gaussian smoothing inequality, which enables the class of functionals to be greatly expanded -- examples are Lipschitz functionals with respect to the uniform metric, and indicators of arbitrary events -- in exchange for a loss of precision in the bounds. Our inequalities are expressed in terms of the smooth test function bound, an expectation of a functional of $X$ that is closely related to classical tightness criteria, a similar expectation for $Z$, and, for the indicator of a set $K$, the probability $\mathbb{P}(Z \in K^\theta \setminus K^{-\theta})$ that the target process is close to the boundary of $K$.