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Bernoulli sums and Rényi entropy inequalities

Mokshay Madiman, James Melbourne, Cyril Roberto

TL;DR

It is proved that a discrete ``min-entropy power'' is super additive on independent variables up to a universal constant, and new bounds on an entropic generalization of the Littlewood-Offord problem that are sharp in the ``Poisson regime''.

Abstract

We investigate the Rényi entropy of independent sums of integer valued random variables through Fourier theoretic means, and give sharp comparisons between the variance and the Rényi entropy, for Poisson-Bernoulli variables. As applications we prove that a discrete ``min-entropy power'' is super additive on independent variables up to a universal constant, and give new bounds on an entropic generalization of the Littlewood-Offord problem that are sharp in the ``Poisson regime''.

Bernoulli sums and Rényi entropy inequalities

TL;DR

It is proved that a discrete ``min-entropy power'' is super additive on independent variables up to a universal constant, and new bounds on an entropic generalization of the Littlewood-Offord problem that are sharp in the ``Poisson regime''.

Abstract

We investigate the Rényi entropy of independent sums of integer valued random variables through Fourier theoretic means, and give sharp comparisons between the variance and the Rényi entropy, for Poisson-Bernoulli variables. As applications we prove that a discrete ``min-entropy power'' is super additive on independent variables up to a universal constant, and give new bounds on an entropic generalization of the Littlewood-Offord problem that are sharp in the ``Poisson regime''.

Paper Structure

This paper contains 10 sections, 29 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Bernoulli Sums
  3. Rényi Entropy Inequalities
  4. Min-Entropy Power
  5. Extreme points
  6. Rearrangement
  7. Integers
  8. Min Entropy Inequality
  9. Min-Entropy inequalities: Tightenings, and Reversals
  10. Littlewood-Offord Problem

Key Result

Theorem 1.2

For $\alpha \in [1,\infty]$, there exists $c(\alpha) \geq 1/e$ such that, for independent $\mathbb{R}^d$-valued random variables $X_i$, Further, for $\alpha \in [0,1)$, there exists $c(\alpha) >0$, such that if $X_i$ are further assumed to be log-concaveWe recall that an $\mathbb{R}^d$-valued random variable is log-concave when it has a density $f$ such that $t \in [0,1]$ and $x,y \in \mathbb{R}^

Theorems & Definitions (66)

  • Definition 1.1: Rényi Entropy Ren61
  • Theorem 1.2: BC14BC15:1RS16MM19
  • Definition 1.3
  • Definition 1.4: Bernoulli Sum
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Proposition 2.4
  • proof
  • ...and 56 more