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Limiting Probability Measures

Irfan Alam

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.

Limiting Probability Measures

Abstract

The coordinates along any fixed direction(s), of points on the sphere , roughly follow a standard Gaussian distribution as 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 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 and its Gaussian mean are obtained in order to complete the above proof. A review of the requisite nonstandard analysis is provided.

Paper Structure

This paper contains 18 sections, 33 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Summary and motivation of our key results
  3. Structure of the paper
  4. A non-logical crash course in nonstandard analysis
  5. Basic nonstandard analysis
  6. Loeb Measures
  7. A Quick nonstandard proof of Poincaré's theorem
  8. When a Loeb measure matches up with a standard measure on a subspace
  9. Basic facts about surface area measures and their nonstandard counterparts
  10. A nonstandard proof of Poincaré's theorem
  11. On the limiting behavior of a sequence of probability spaces
  12. Integrating finite dimensional functions along nice sequences of probability spaces
  13. Application to a proof of the Riemann-Lebesgue Lemma
  14. What happens if the finite dimensional function is not nice in the limiting space?
  15. Generalizing Poincaré's theorem
  16. ...and 3 more sections

Key Result

Theorem 1.1

For all bounded measurable functions $f: \mathbb{R}^k \rightarrow \mathbb{R}$, we have

Theorems & Definitions (59)

  • Theorem 1.1: Poincaré, Poincare
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • ...and 49 more