  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 $\ell_\psi$. 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:
$\alpha_\psi^0>\beta_M^\infty$.
