On subspaces of Orlicz spaces spanned by independent copies of a mean zero function

Abstract

We study the subspaces of the Orlicz spaces $L_M$ spanned by independent copies $f_k$, $k=1,2,…$, of a function $f\in L_M$, $\int_0^1 f(t) dt=0$. Any such a subspace $H$ is isomorphic to some Orlicz sequence space $\ell_\psi$. In terms of dilations of the function $f$, a description of strongly embedded subspaces of this type is obtained, and conditions guaranteeing that the unit ball of such a subspace consists of functions with equicontinuous norms in $L_M$ are found. In particular, we prove that there is a wide class of Orlicz spaces $L_M$ (containing the $L^p$-spaces, $1\le p< 2$), for which each of the above properties of $H$ holds if and only if the Matuszewska-Orlicz indices of the functions $M$ and $\psi$ satisfy $\alpha_\psi^0>\beta_M^\infty$.