Systems of random variables equivalent in distribution to the Rademacher system and the \(\mathcal K\)-closed representability of Banach couples. (English. Russian original) Zbl 0965.60016

Astashkin, S. V.

Sb. Math. 191, No. 6, 779-807 (2000); translation from Mat. Sb. 191, No. 6, 3-30 (2000).

A problem of selecting a subsequence, which is equivalent in distribution to the Rademacher system, from a given sequence of random variables is considered. Necessary and sufficient conditions for the existence of such subsequence are given. At first a theorem which expresses the equivalence in distribution to the Rademacher system in terms of \(L_p\)-norms and in terms of the Petre \({\mathcal K}\)-functionals is given. It is proved that a uniformly bounded strongly multiplicative system \((f_n)_n\) of random variables such that \(\inf_n Ef^2_n>0\), is equivalent in distribution to the Rademacher system. Next, a problem of selection of a subsystem, which is equivalent in distribution to the Rademacher system is solved. In particular, it is always possible to extract from a uniformly bounded orthonormal infinite system \((g_n)_n\) of random variables an infinite subsystem equivalent in distribution to the Rademacher system. From a uniformly bounded orthonormal finite system \((g_n)^N_{n=1}\) we can select a subsystem of logarithmic density (i.e. a subsystem which contains at least \(C\log_2N\) functions). \({\mathcal K}\)-closed representability of Banach couples is also considered.

Reviewer: J.Jakubowski (Warszawa)

MSC:

60B12	Limit theorems for vector-valued random variables (infinite-dimensional case)
28A20	Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence

Keywords:

Rademacher system; equivalence in distribution