Exponential inequalities in probability spaces revisited

TL;DR

This paper revisits and extends exponential inequalities in probability spaces, providing new quantitative forms and dual arguments applicable to continuous and discrete measures, including Gaussian and Poisson measures.

Contribution

It introduces a novel dual argument approach to exponential inequalities, extending recent results and deriving new inequalities in both continuous and discrete probability measures.

Findings

Quantitative form of Moser-Trudinger-type inequalities

Recovery of Ivanisvili-Russell's inequality for Gaussian measure

Extension of inequalities to Poisson measure on integers

Abstract

We revisit several results on exponential integrability in probability spaces and derive some new ones. In particular, we give a quantitative form of recent results by Cianchi-Musil and Pick in the framework of Moser-Trudinger-type inequalities, and recover Ivanisvili-Russell's inequality for the Gaussian measure. One key ingredient is the use of a dual argument, which is new in this context, that we also implement in the discrete setting of the Poisson measure on integers.