Hölder's inequality and its reverse-a probabilistic point of view

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 ${\mathbb{R}}^n$. 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 ${\mathrm{\ell }}_p^n$ balls and spheres. We also provide a Berry-Esseen type result and prove a large and a moderate deviation principle for the suitably normalized Hölder ratio.