Lacunary Series, Nonlinear Functionals and Banach Space Structure

In a previous paper \cite{BT} we studied the asymptotic behavior of $\mathrm{\parallel }{\sum }_{k=1}^N{a}_k{X}_{n_k}{\mathrm{\parallel }}_p$ for lacunary sequences $(X_{n_k})$ of random variables in $L_p$ and used the result to give a necessary and sufficient condition for the first alternative in the Kadec-Pełczynski theorem in the case $1\le p<2$. In the present paper we extend this result for nonlinear functionals $f_k(a_1{X}_{n_1},\dots ,a_k{X}_{n_k})$, establishing a uniform version of the subsequence principle of Aldous \cite{ald}. Moreover, we prove Kadec-Pełczynski type theorems in Orlicz spaces $L_{\psi }$.