Sets of random variables with a given uncorrelation structure (Q1612943)

Let \(\xi_1,\dots,\xi_n\) be random variables having finite expectations. Denote \[ i_k:=\# \left\{(j_1,\dots, j_k):1\leq j_1<\cdots <j_k\leq n\text{ and }{\mathbf E}\prod^k_{l=1}\xi_{j_l}= \prod^k_{l=1}{\mathbf E}\xi_{j_l}\right\},\;k=2,\dots,n. \] The finite sequence \((i_2,\dots,i_n)\) is called the uncorrelation structure of \(\xi_1,\dots, \xi_n\). It is proved that for any given sequence of nonnegative integers \((i_2,\dots, i_n)\) satisfying \(0\leq i_k\leq {n\choose k}\) and any given nondegenerate probability distributions \(P_1,\dots,P_n\) there exist random variables \(\eta_1,\dots, \eta_n\) with respective distributions \(P_1, \dots,P_n\) such that \((i_2,\dots,i_n)\) is their uncorrelation structure.