On the Distribution of Random variables corresponding to Musielak-Orlicz 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 $(x_i{X}_i{)}_{i=1}^n$ is of the order $\Vert x{\Vert }_M$ (up to constants dependent on p only). In case $p=2$ we need the function $t\mapsto t{M}^{\prime }(t)-M(t)$ 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, Schütt, 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.

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