Sets of random variables with a given uncorrelation structure

Let [xi]1,...,[xi]n be random variables having finite expectations. DenoteThe finite sequence (i2,...,in) is called the uncorrelation structure of [xi]1,...,[xi]n. It is proved that for any given sequence of nonnegative integers (i2,...,in) satisfying and any given nondegenerate probability distributions P1,...,Pn there exist random variables [eta]1,...,[eta]n with respective distributions P1,...,Pn such that (i2,...,in) is their uncorrelation structure.