Ruc property for chaos of random variables in the uniform norm

Let $X=\{X_k\}_{k=1}^\infty$ be a sequence of independent symmetric bounded random variables. This paper investigates systems of the form $\{X_iX_j\}_{i<j}$, $\{X_i X_j X_k\}_{i<j<k},\ldots$, finite unions of such systems, and systems close to them, in the space $L_\infty$ of bounded random variables. Series over such systems do not hold the property of unconditionality: the convergence of the series depends on the ordering of the terms. At the same time, as we demonstrate in the paper, such systems posess a very close property of random unconditional convergence (or RUC-property).