An Inversion Formula for Orlicz Norms and Sequences of Random Variables

TL;DR

This paper establishes an inversion formula linking Orlicz functions to the distribution of random variables that generate the associated Orlicz norms, providing a practical way to understand their structure.

Contribution

It introduces a method to identify which random variables produce a given Orlicz norm and offers a distribution function representation based on the Orlicz function and its derivative.

Findings

Derived an inversion formula connecting Orlicz functions and random variables.

Provided a distribution function representation in terms of M and M' for practical applications.

Enhanced understanding of the structure of random variables generating Orlicz norms.

Abstract

Given an Orlicz function $M$, we show which random variables $\xi_i$, $i=1,...,n$ generate the associated Orlicz norm, i.e., which random variables yield $\mathbb{E} \max\limits_{1\leq i \leq n}|x_i\xi_i| \sim \norm{(x_i)_{i=1}^n}_M$. As a corollary we obtain a representation for the distribution function in terms of $M$ and $M'$ which can be easily applied to many examples of interest.