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arXiv · 1305.1442

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

David Alonso-Gutierrez·Soeren Christensen·Markus Passenbrunner·Joscha Prochno
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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, 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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David Alonso-Gutierrez, Soeren Christensen, Markus Passenbrunner, Joscha Prochno. 2013-05-07. On the Distribution of Random variables corresponding to Musielak-Orlicz norms. https://doi.org/10.4064/sm219-3-6

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