Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Logarithmic Sobolev inequalities for generalised Cauchy measures

Baptiste Nicolas Huguet

Abstract

We prove a curvature-dimension criterion and obtain logarithmic Sobolev inequalities for generalised Cauchy measures with optimal weights and explicit constants. In the one-dimensional case, this constant is even optimal. From these inequalities, we derive concentration results, which allow concluding the case of the pathological dimension two.

Logarithmic Sobolev inequalities for generalised Cauchy measures

Abstract

We prove a curvature-dimension criterion and obtain logarithmic Sobolev inequalities for generalised Cauchy measures with optimal weights and explicit constants. In the one-dimensional case, this constant is even optimal. From these inequalities, we derive concentration results, which allow concluding the case of the pathological dimension two.

Paper Structure

This paper contains 5 sections, 13 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Rescaled generalised Cauchy measures
  3. One-dimensional case
  4. General case
  5. Two-dimensional case: a tensorisation argument

Key Result

Lemma 2.1

Assume that $\mu_{\beta,\sigma}$ satisfies a logarithmic Sobolev inequality with weight $\omega_{\sigma}\log(\omega_{\sigma})$ and constant $C>0$, then $\mu_\beta$ satisfies a logarithmic Sobolev inequality with weight $\omega(\log(\sigma) + \log(\omega))$ and the same constant $C$. $\blacktrianglel

Theorems & Definitions (23)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3: ChL
  • Proposition 2.4
  • proof
  • Proposition 3.1
  • Theorem 3.2
  • proof
  • ...and 13 more