Sparse Rademacher chaos in symmetric spaces

Abstract:

A dispersed Rademacher chaos whose combinatorial dimension equals its order $d$ is treated. It is proved that its unconditionality in a symmetric space $X$ guarantees the equivalence of this chaos in $X$ to the canonical basis of $\ell _2$. In its turn, the latter property occurs if and only if $X\supset G_{2/d}$, where $G_{2/d}$ is the separable part of the Orlicz space $\mathrm {ExpL}^{2/d}$ corresponding to the function $M(u)\sim \mathrm {exp}(u^{2/d})$. Furthermore, it is shown that a chaos of an arbitrary order constructed by an arbitrary system of stochastically independent symmetric random variables is a basic sequence in any ambient symmetric space.