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 X₁,...,Xₙ are independent copies of X, then , where is a positive constant depending only on p. In case p = 2 we need the function t ↦ tM’(t) - M(t) to be 2-concave and as an application immediately obtain an embedding of the corresponding Orlicz spaces into L₁[0,1]. We also provide a general result replacing the -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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm219-3-6,
author = {David Alonso-Guti\'errez and S\"oren Christensen and Markus Passenbrunner and Joscha Prochno},
title = {On the distribution of random variables corresponding to Musielak-Orlicz norms},
journal = {Studia Mathematica},
volume = {215},
year = {2013},
pages = {269-287},
zbl = {1318.46005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm219-3-6}
}
David Alonso-Gutiérrez; Sören Christensen; Markus Passenbrunner; Joscha Prochno. On the distribution of random variables corresponding to Musielak-Orlicz norms. Studia Mathematica, Tome 215 (2013) pp. 269-287. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm219-3-6/