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Rigid characterizations of probability measures through independence, with applications

Thomas A. Courtade

Abstract

Three equivalent characterizations of probability measures through independence criteria are given. These characterizations lead to a family of Brascamp--Lieb-type inequalities for relative entropy, determine equilibrium states and sharp rates of convergence for certain linear Boltzmann-type dynamics, and unify an assortment of $L^2$ inequalities in probability.

Rigid characterizations of probability measures through independence, with applications

Abstract

Three equivalent characterizations of probability measures through independence criteria are given. These characterizations lead to a family of Brascamp--Lieb-type inequalities for relative entropy, determine equilibrium states and sharp rates of convergence for certain linear Boltzmann-type dynamics, and unify an assortment of inequalities in probability.
Paper Structure (12 sections, 13 theorems, 106 equations)

This paper contains 12 sections, 13 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Characterizations of measures through independence
  3. Relationship to other work
  4. Applications
  5. Bernstein-type characterization of Gaussians
  6. Geometric Brascamp--Lieb-type inequalities
  7. Linear Boltzmann-type dynamics
  8. Inequalities in probability
  9. Proof of Theorem \ref{['thm:splittingwrtxi']}
  10. Special case of Theorem \ref{['thm:splittingwrtxi']} when $\xi$ is a discrete measure
  11. Proof of Theorem \ref{['thm:splittingwrtxi']} in the general case
  12. Remarks on Question \ref{['q:moments']}

Key Result

Proposition 1

The set $\chi_{\xi}^{-1}(0)\subset \mathbb{R}^n$ may be uniquely written as the union of mutually orthogonal, distinct linear subspaces $(E_{\alpha})_{\alpha}\subset \mathbf{E}$, which we denote

Theorems & Definitions (41)

  • Definition 1
  • Proposition 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • Proposition 2
  • ...and 31 more