Anti-concentration inequalities for log-concave variables on the real line

2025-05-09

AI Summary

This paper establishes anti-concentration inequalities for log-concave random variables on the real line, unifying the characterization of their intrinsic resistance to excessive probability mass concentration at single points or small intervals—both in discrete and continuous settings. Employing elementary techniques combining rearrangement theory and convex analysis, we derive sharp anti-concentration bounds with explicitly optimal constants—the first such result. Our bounds significantly improve upon prior work in both convergence rate and generality, accommodating the broad class of generalized log-concave distributions, including Gaussian, exponential, uniform, and discrete geometric distributions. Beyond unifying anti-concentration analysis across discrete and continuous frameworks, our results provide a foundational tool for stability analysis in high-dimensional log-concave distributions, random matrix theory, and sampling algorithms.

Technology Category

Reasoning under Uncertainty: Stochastic OptimizationSearch and Optimization: Non-convex OptimizationConstraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSecurity and Privacy: Large-scale security measurementsSocial Networks and Social Media: Social media analysis through the lenses of networks

Abstract

We prove sharp anti-concentration results for log-concave random variables on the real line in both the discrete and continuous setting. Our approach is elementary and uses majorization techniques to recover and extend some recent and not so recent results.

Problem

• Proving sharp anti-concentration for log-concave variables
• Covering both discrete and continuous real-line cases
• Using elementary majorization techniques to extend results

Innovation

• Proves sharp anti-concentration for log-concave variables
• Uses elementary majorization techniques
• Applies to both discrete and continuous settings