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

David Alonso-Gutiérrez, et al. "On the distribution of random variables corresponding to Musielak-Orlicz norms." Studia Mathematica 219.3 (2013): 269-287. <http://eudml.org/doc/285411>.

@article{DavidAlonso2013,
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 X₁,...,Xₙ are independent copies of X, then $1/C_\{p\} ||x||_M ≤ ||(x_\{i\}X_\{i\})ⁿ_\{i=1\}||_\{p\} ≤ C_\{p\}||x||_M$, where $C_\{p\}$ 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 $ℓ_\{p\}$-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.},
author = {David Alonso-Gutiérrez, Sören Christensen, Markus Passenbrunner, Joscha Prochno},
journal = {Studia Mathematica},
keywords = {Orlicz function; Orlicz norm; Musielak-Orlicz norm; random variable; distribution},
language = {eng},
number = {3},
pages = {269-287},
title = {On the distribution of random variables corresponding to Musielak-Orlicz norms},
url = {http://eudml.org/doc/285411},
volume = {219},
year = {2013},
}

TY - JOUR
AU - David Alonso-Gutiérrez
AU - Sören Christensen
AU - Markus Passenbrunner
AU - Joscha Prochno
TI - On the distribution of random variables corresponding to Musielak-Orlicz norms
JO - Studia Mathematica
PY - 2013
VL - 219
IS - 3
SP - 269
EP - 287
AB - 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 $1/C_{p} ||x||_M ≤ ||(x_{i}X_{i})ⁿ_{i=1}||_{p} ≤ C_{p}||x||_M$, where $C_{p}$ 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 $ℓ_{p}$-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.
LA - eng
KW - Orlicz function; Orlicz norm; Musielak-Orlicz norm; random variable; distribution
UR - http://eudml.org/doc/285411
ER -