On norms in some class of exponential type Orlicz spaces of random variables

TL;DR

This paper introduces a new way to characterize exponential type Orlicz spaces of random variables, defining equivalent norms and demonstrating their application through probabilistic inequalities like Hoeffding's.

Contribution

It provides a novel characterization and norm equivalence for exponential type Orlicz spaces, with applications to probabilistic inequalities.

Findings

Defined new norms for centered random variables in these spaces

Proved equivalence of the new norms with Luxemburg norms

Applied the norms to Hoeffding inequality in probability theory

Abstract

A new characterization of the exponential type Orlicz spaces generated by the functions $\exp(|x|^p)-1$ ($p\ge 1$) is given. We define norms for centered random variables belonging to these spaces. We show equivalence of these norms with the Luxemburg norms. On the example of the Hoeffding inequality we present some application of these norms in a probabilistic context.