Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Transport inequalities for random point measures

Nathael Gozlan, Ronan Herry, Giovanni Peccati

Abstract

We derive transport-entropy inequalities for mixed binomial point processes, and for Poisson point processes. We show that when the finite intensity measure satisfies a Talagrand transport inequality, the law of the point process also satisfies a Talagrand type transport inequality. We also show that a Poisson point process (with arbitrary $σ$-finite intensity measure) always satisfies a universal transport-entropy inequality à la Marton. We explore the consequences of these inequalities in terms of concentration of measure and modified logarithmic Sobolev inequalities. In particular, our results allow one to extend a deviation inequality by Reitzner [31], originally proved for Poisson random measures with finite mass.

Transport inequalities for random point measures

Abstract

We derive transport-entropy inequalities for mixed binomial point processes, and for Poisson point processes. We show that when the finite intensity measure satisfies a Talagrand transport inequality, the law of the point process also satisfies a Talagrand type transport inequality. We also show that a Poisson point process (with arbitrary -finite intensity measure) always satisfies a universal transport-entropy inequality à la Marton. We explore the consequences of these inequalities in terms of concentration of measure and modified logarithmic Sobolev inequalities. In particular, our results allow one to extend a deviation inequality by Reitzner [31], originally proved for Poisson random measures with finite mass.

Paper Structure

This paper contains 33 sections, 26 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on measures and optimal transport
  3. Topology of spaces of measures
  4. Optimal transport between measures of arbitrary masses
  5. Couplings, transport costs and generalized transport costs
  6. Some examples
  7. Monge-Kantorovich distance
  8. Marton-type transport distances
  9. Existence of optimal couplings, lower semicontinuity of transport costs and stability of optimal couplings
  10. Optimal (partial) transport between configurations
  11. Transport-entropy inequalities
  12. Definitions and examples
  13. Talagrand's inequality
  14. Marton's universal transport inequalities
  15. Stability by tensorization, by push-forward, and by approximation
  16. ...and 18 more sections

Key Result

Theorem \oldthetheorem

Suppose that $\mu \in \mathcal{P}(Z)$ satisfies Talagrand's inequality eq:Talagrand with a constant $a>0$, then for any $\kappa \in \mathcal{P}(\mathbb{N})$, the probability measure $B_{\mu,\kappa}\in \mathcal{P}(\mathcal{M}_b(Z))$ satisfies the following inequality: for all $\Pi_1,\Pi_2 \in \mathca where $\lambda \in \mathcal{P}(\mathbb{R}_+)$ is such that for all $i\in \{1,2\}$, $\Pi_i( \{\eta \

Theorems & Definitions (46)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • ...and 36 more