Khinchin-type inequalities via Hadamard's factorisation

TL;DR

This paper establishes sharp Khinchin-type inequalities for type L random variables using complex analysis and symmetric function inequalities, extending results to dependent variables and comparing related probabilistic notions.

Contribution

It introduces new sharp inequalities for type L variables, including dependent cases, and clarifies the relationships among ultra sub-Gaussianity, strong log-concavity, and type L.

Findings

Sharp Khinchin inequalities with optimal constants for type L variables

Extension to dependent variables with ferromagnetic dependencies

Equivalence of ultra sub-Gaussianity and strong log-concavity

Abstract

We prove Khinchin-type inequalities with sharp constants for type L random variables and all even moments. Our main tool is Hadamard's factorisation theorem from complex analysis, combined with Newton's inequalities for elementary symmetric functions. Besides the case of independent summands, we also treat ferromagnetic dependencies in a nonnegative external magnetic field (thanks to Newman's generalisation of the Lee-Yang theorem). Lastly, we compare the notions of type L, ultra sub-Gaussianity (introduced by Nayar and Oleszkiewicz) and strong log-concavity (introduced by Gurvits), with the latter two being equivalent.