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Concentration inequalities for log-concave sequences

Arnaud Marsiglietti, James Melbourne

TL;DR

The paper investigates how quantitative aspects of log-concavity manifest in probability by employing convex majorization and relative log-concavity. It develops a general convex-domination framework that compares log-concave variables to extremal log-affine (and Gamma) references, enabling transfer of Chernoff-type tails, moment inequalities, and Rényi-entropy bounds. Key contributions include a broad extension of convex majorization, sharp concentration for discrete log-concave sums, Gaussian-type bounds for ultra log-concave structures, Poisson-type bounds for intrinsic-volume distributions, and precise entropy-maximization results; it also clarifies the behavior of intrinsic volumes and provides continuous analogs. Overall, the work unifies discrete and continuous log-concavity theories, sharpens existing bounds, and yields powerful tools for applications in combinatorics, geometry, and beyond.

Abstract

We investigate quantitative implications of the notion of log-concavity through a probabilistic interpretation. In particular, we derive concentration inequalities, moment and entropy bounds for random variables satisfying a precise degree of log-concavity. Along the way, we recover, improve, and simplify several results existing in the literature. Our approach is based on majorization in the convex order.

Concentration inequalities for log-concave sequences

TL;DR

The paper investigates how quantitative aspects of log-concavity manifest in probability by employing convex majorization and relative log-concavity. It develops a general convex-domination framework that compares log-concave variables to extremal log-affine (and Gamma) references, enabling transfer of Chernoff-type tails, moment inequalities, and Rényi-entropy bounds. Key contributions include a broad extension of convex majorization, sharp concentration for discrete log-concave sums, Gaussian-type bounds for ultra log-concave structures, Poisson-type bounds for intrinsic-volume distributions, and precise entropy-maximization results; it also clarifies the behavior of intrinsic volumes and provides continuous analogs. Overall, the work unifies discrete and continuous log-concavity theories, sharpens existing bounds, and yields powerful tools for applications in combinatorics, geometry, and beyond.

Abstract

We investigate quantitative implications of the notion of log-concavity through a probabilistic interpretation. In particular, we derive concentration inequalities, moment and entropy bounds for random variables satisfying a precise degree of log-concavity. Along the way, we recover, improve, and simplify several results existing in the literature. Our approach is based on majorization in the convex order.
Paper Structure (11 sections, 30 theorems, 141 equations, 1 figure)

This paper contains 11 sections, 30 theorems, 141 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background on Convex Majorization
  3. Majorization of Log-Concave Distributions
  4. Quantitative implications
  5. Concentration inequalities
  6. Moment Bounds
  7. Maximum Entropy Distributions
  8. Log-concave Sequences
  9. Ultra log-concavity of order $n$ and ULC
  10. Log-concavity of intrinsic volumes
  11. Continuous setting

Key Result

Theorem 2.2

Let $\mu$ be a measure on $[0, +\infty)$. Suppose that $f$ and $g$ are Borel measurable functions on $[0, +\infty)$ such that and and the integrals are finite. If there exists an interval $I \coloneqq [a,b] \subseteq (0,\infty)$ such that $g(y) \leq f(y)$ for $y \in I$ while $g(y) \geq f(y)$ for $y \notin I$, then $\varphi \colon (0,\infty) \to \mathbb{R}$ convex implies,

Figures (1)

  • Figure 1: Relations between log-concavity notions

Theorems & Definitions (65)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • ...and 55 more