Systems of random variables equivalent in distribution to the Rademacher system and K-closed representability of Banach couples

Necessary and sufficient conditions ensuring that one can select from a system {f(n)}(n=1)(infinity) of random variables on a probability space (Omega, Sigma, P) a subsystem {phi(i)}(i=1)(infinity) equivalent in distribution to the Rademacher system on [0, 1] are found. In particular, this is always possible if {f(n)}(n=1)(infinity) is a uniformly bounded orthonormal sequence. The main role in the proof is played by the connection (discovered in this paper) between the equivalence in distribution of random variables and the behaviour of the L-p-norms of the corresponding polynomials. An application of the results obtained to the study of the K-closed representability of Banach couples is presented.