This is an announcement for the paper "On the Distribution of Random variables corresponding to norms" by David Alonso-Gutierrez, Soeren Christensen, Markus Passenbrunner, and Joscha Prochno.
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.
Archive classification: math.FA math.PR
Mathematics Subject Classification: 46B09, 46B07, 46B45, 60B99
Submitted from: joscha.prochno@jku.at
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1305.1442
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