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Limiting spherical integrals of bounded continuous functions

Irfan Alam

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 $μ$ 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.

Limiting spherical integrals of bounded continuous functions

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 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 intersected with certain affine subspaces of . 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.

Paper Structure

This paper contains 11 sections, 20 theorems, 92 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation and definitions
  3. Description of key results
  4. Integrating bounded functions on certain great circles
  5. A hyperfinite approximation and integrating on any great circle
  6. Effect of an infinitesimal rotation on the ${^*}$-integral of a bounded uniformly continuous function
  7. Some integral continuity properties
  8. A hyperfinite approximation via overflow
  9. Integrating continuous functions over non-great circles
  10. Some results on linear independence
  11. Working with infinitesimally separated linear spaces

Key Result

Theorem 1.1

Let $f\colon \mathbb{R}^k \rightarrow \mathbb{R}$ be bounded and Borel measurable. If the Gaussian measure ${\mu}_{\bar{\eta}, u^{(1)}, \ldots, u^{(\gamma)}}$ has full support on $\mathbb{R}^k$, then

Figures (3)

  • Figure 1: Intersecting $S^{n-1}(\sqrt{n})$ by the affine plane $A_n$
  • Figure 2: $S^{(1)}$ and $S^{(2)}$ are separated infinitesimally
  • Figure 3: Visualizing $S_{A_N}$ in contrast with $S_{H_N}$

Theorems & Definitions (46)

  • Theorem 1.1: Peterson--Sengupta
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 36 more