An Inversion Formula for Orlicz Norms and Sequences of Random Variables

Given an Orlicz function $M$, we show which random variables ${\xi }_i$, $i=1,\mathrm{.}\mathrm{.}\mathrm{.},n$ 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 $M^{\mathrm{\prime }}$ which can be easily applied to many examples of interest.