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

We study subspaces of Orlicz spaces $L_M$ spanned by independent copies $f_k$, $k=1,2,\dots$, 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 ${\mathrm{\ell }}_{\psi }$. 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 $L^p$-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 the inequality: ${\alpha }_{\psi }^0>{\beta }_M^{\mathrm{\infty }}$.