  We present necessary and sufficient conditions on systems of random variables
for them to possess a lacunary subsystem equivalent in distribution to the
Rademacher system on the segment [0,1]. In particular, every uniformly bounded
orthonormal system has this property. Furthermore, an arbitrary finite
uniformly bounded orthonormal set of N functions contains a subset of
"logarithmic" density equivalent in distribution to the corresponding set of
Rademacher functions, with a constant independent of N. A connection between
the tail distribution and the L_p-norms of polynomials with respect to systems
of random variables exploited. We use also these results to study K-closed
representability of some Banach couples.
