Hölder's inequality and its reverse—A probabilistic point of view

Abstract

In this article, we take a probabilistic look at Hölder's inequality, considering the ratio of terms in the classical Hölder inequality for random vectors in (Formula presented.). 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 (Formula presented.) balls and spheres. We also provide a Berry–Esseen–type result and prove a large and a moderate deviation principle for the suitably normalized Hölder ratio.