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## On the distribution of random variables corresponding to Musielak–Orlicz norms

### Tom 219 / 2013

Studia Mathematica 219 (2013), 269-287 MSC: Primary 46B09; Secondary 46B07, 46B45, 60B99. DOI: 10.4064/sm219-3-6

#### Streszczenie

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 $X_1,\ldots,X_n$ are independent copies of $X$, then $\frac{1}{C_p} \|x\|_M \leq \mathbb E \|(x_iX_i)_{i=1}^n\|_p \leq C_p\|x\|_M,$ where $C_p$ is a positive constant depending only on $p$. 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 [Ann. Prob. 30 (2002)]. We also prove, in the spirit of that paper, a result which is of a simpler form and easier to apply. All results are true in the more general setting of Musielak–Orlicz spaces.

#### Autorzy

• David Alonso-GutiérrezDepartamento de Matemáticas
Campus de Espinardo
30100 Murcia, Spain
and
Departament de Matemàtiques
Universitat Jaume I
Campus de Riu Sec
E-12071 Castelló de la Plana, Spain
e-mail
• Sören ChristensenMathematisches Seminar
Christian Albrechts University Kiel
Ludewig-Meyn-Strasse 4
24098 Kiel, Germany
and
Department of Mathematics
SPST, University of Hamburg
Bundesstrasse 55
20146 Hamburg, Germany
e-mail
• Markus PassenbrunnerInstitute of Analysis
Johannes Kepler University Linz
Altenbergerstrasse 69
4040 Linz, Austria
e-mail
• Joscha ProchnoInstitute of Analysis
Johannes Kepler University Linz
Altenbergerstrasse 69
4040 Linz, Austria
e-mail

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