# Limiting Probability Measures

## TL;DR

This paper offers a nonstandard analysis approach to classical high-dimensional probability results, providing new proofs and a unified theory for asymptotic integrals over spheres and Gaussian measures.

## Contribution

It introduces a nonstandard framework for analyzing asymptotic behaviors of integrals on high-dimensional spheres, including new proofs of classical results and the Riemann--Lebesgue lemma.

## Key findings

- New proof of the Gaussian distribution of coordinates on high-dimensional spheres
- Development of a nonstandard theory for asymptotic integrals over varying domains
- Establishment of inequalities relating spherical means and Gaussian means

## Abstract

The coordinates along any fixed direction(s), of points on the sphere $S^{n-1}(\sqrt{n})$, roughly follow a standard Gaussian distribution as $n$ approaches infinity. We revisit this classical result from a nonstandard analysis perspective, providing a new proof by working with hyperfinite dimensional spheres. We also set up a nonstandard theory for the asymptotic behavior of integrals over varying domains in general. We obtain a new proof of the Riemann--Lebesgue lemma as a by-product of this theory. We finally show that for any function $f \co \mathbb{R}^k \to \mathbb{R}$ with finite Gaussian moment of an order larger than one, its expectation is given by a Loeb integral integral over a hyperfinite dimensional sphere. Some useful inequalities between high-dimensional spherical means of $f$ and its Gaussian mean are obtained in order to complete the above proof. A review of the requisite nonstandard analysis is provided.