On the Distribution of Random variables corresponding to Musielak-Orlicz norms

TL;DR

This paper establishes a formula for the distribution of random variables linked to Musielak-Orlicz norms, enabling the embedding of these spaces into L1 and generalizing previous results.

Contribution

It introduces an explicit distribution formula for Musielak-Orlicz norms and extends embedding results to more general Musielak-Orlicz spaces.

Findings

Derived a distribution formula for Musielak-Orlicz norms.

Established embedding of Musielak-Orlicz spaces into L1.

Generalized results to arbitrary N-norms.

Abstract

Given a normalized Orlicz function $M$ we provide an easy formula for a distribution such that, if $X$ is a random variable distributed accordingly and $X_1,...,X_n$ are independent copies of $X$, then the expected value of the p-norm of the vector $(x_iX_i)_{i=1}^n$ is of the order $\| x \|_M$ (up to constants dependent on p only). In case $p=2$ we need the function $t\mapsto tM'(t) - M(t)$ to be 2-concave and as an application immediately obtain an embedding of the corresponding Orlicz spaces into $L_1[0,1]$. We also provide a general result replacing the $\ell_p$-norm by an arbitrary $N$-norm. This complements some deep results obtained by Gordon, Litvak, Sch\"utt, and Werner. We also prove a result in the spirit of their work which is of a simpler form and easier to apply. All results are true in the more general setting of Musielak-Orlicz spaces.