2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/59685We 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.26 pagesFunctional AnalysisClassical Analysis and ODEs42A55 (primary), 42A61,46B70 (secondary)text