Limiting spherical integrals of bounded continuous functions

TL;DR

This paper employs nonstandard analysis to extend the understanding of how Gaussian integrals can be approximated by spherical integrals, even when the Gaussian measure lacks full support, providing a new geometric perspective.

Contribution

It generalizes previous results by using nonstandard analysis to handle Gaussian measures without full support through Loeb integrals over hyperfinite spheres.

Findings

Extended the limiting behavior of spherical integrals to non-fully supported Gaussian measures.

Provided a nonstandard geometric framework for analyzing Gaussian Radon transforms.

Included an asymptotic linear algebra result supporting the main analysis.

Abstract

We use nonstandard analysis to study the problem of expressing a Gaussian integral in terms of the limiting behavior of a sequence of spherical integrals. Peterson and Sengupta proved that if a Gaussian measure $\mu$ has full support on a finite-dimensional Euclidean space, then the expected value of a bounded measurable function on that domain can be expressed as a limit of integrals over spheres $S^{n-1}(\sqrt{n})$ intersected with certain affine subspaces of $\mathbb{R}^n$. This allows one to realize the Gaussian Radon transform of such functions as a limit of spherical integrals. Using nonstandard analysis, we study such limits in terms of Loeb integrals over a single hyperfinite dimensional sphere. This nonstandard geometric approach generalizes the known limiting result for bounded continuous functions to the case when the Gaussian measure is not necessarily fully supported. We also present an asymptotic linear algebra result needed in the above proof.