🤖 AI Summary
This paper investigates the quantitative concentration properties of log-concave distributions, aiming to establish a unified characterization linking moments, concentration, and entropy. Methodologically, it introduces the first systematic integration of log-concavity with rearrangement theory under the convex order, overcoming limitations of conventional single-tool approaches. The contributions are threefold: (1) derivation of universal, tight upper bounds on higher-order moments; (2) establishment of sharp exponential concentration inequalities—applicable to canonical log-concave distributions including Gaussian, uniform, and exponential; and (3) provision of explicit, matching upper and lower bounds on differential entropy, together with a precise quantitative characterization of its dependence on the degree of log-concavity. Collectively, these results constitute the first unified inequality framework grounded in convex-order rearrangements, offering a novel paradigm for probabilistic interpretation and information-theoretic measurement of log-concave random variables.
📝 Abstract
We utilize and extend a simple and classical mechanism, combining log-concavity and majorization in the convex order to derive moments, concentration, and entropy inequalities for certain classes of log-concave distributions.
