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

David Alonso-Gutiérrez¹ ; Sören Christensen² ; Markus Passenbrunner³ ; Joscha Prochno³

¹ Departamento de Matemáticas Universidad de Murcia Campus de Espinardo 30100 Murcia, Spain and Departament de Matemàtiques Universitat Jaume I Campus de Riu Sec E-12071 Castelló de la Plana, Spain
² Mathematisches 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
³ Institute of Analysis Johannes Kepler University Linz Altenbergerstrasse 69 4040 Linz, Austria

Studia Mathematica, Tome 219 (2013) no. 3, pp. 269-287

Résumé

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.

Détail

DOI : 10.4064/sm219-3-6

Keywords: given normalized orlicz function provide easy formula distribution random variable distributed accordingly ldots independent copies frac leq mathbb leq where positive constant depending only function mapsto concave application immediately obtain embedding corresponding orlicz spaces provide general result replacing ell p norm arbitrary n norm complements deep results obtained gordon litvak sch werner ann prob prove spirit paper result which simpler form easier apply results general setting musielak orlicz spaces