Negative Moments of Steinhaus Sums

TL;DR

This paper establishes a precise upper bound on negative moments of sums of independent Steinhaus variables, completing the characterization of sharp inequalities for these sums and correcting previous inaccuracies.

Contribution

It provides a sharp upper bound on negative moments of Steinhaus sums and corrects a mistake in earlier literature, advancing the understanding of Khinchin-type inequalities.

Findings

Sharp upper bound on negative moments of Steinhaus sums

Correction of a previous mistake in related literature

Application to bounds on Rényi entropy

Abstract

We prove a sharp upper bound on negative moments of sums of independent Steinhaus random variables (that is uniform on circles in the plane). Together with the series of earlier works: K\"onig-Kwapie\'n (2001), Baernstein II-Culverhouse (2002), and K\"onig (2014), this closes the investigation of sharp $L_p-L_2$ Khinchin-type inequalities for the Steinhaus sums. Incidentally, we fix a mistake in an earlier paper, as well as provide an application to sharp bounds on R\'enyi entropy.