H\"older's inequality and its reverse-a probabilistic point of view

TL;DR

This paper investigates H"older's inequality from a probabilistic perspective, establishing a central limit theorem and deviation principles for the ratio of terms in the inequality for random vectors, with applications to high-dimensional probability models.

Contribution

It introduces a probabilistic framework for H"older's inequality, proving a CLT, deviation principles, and reverse inequalities for random vectors in high-dimensional spaces.

Findings

Established a CLT for the H"older ratio in high dimensions

Proved a large deviation principle for the ratio

Demonstrated a reverse inequality with high probability

Abstract

In this article we take a probabilistic look at H\"older's inequality, considering the ratio of terms in the classical H\"older inequality for random vectors in $\mathbb{R}^n$. We prove a central limit theorem for this ratio, which then allows us to reverse the inequality up to a multiplicative constant with high probability. The models of randomness include the uniform distribution on $\ell_p^n$ balls and spheres. We also provide a Berry-Esseen type result and prove a large and a moderate deviation principle for the suitably normalized H\"older ratio.