Sets of random variables with a given uncorrelation structure

Abstract

Let ξ 1,…, ξ n be random variables having finite expectations. Denote i k ≔ # (j 1,…,j k) : 1⩽j 1<⋯<j k⩽n and E ∏ l=1 k ξ j l = ∏ l=1 k E ξ j l , k=2,…,n. The finite sequence ( i 2,…, i n ) is called the uncorrelation structure of ξ 1,…, ξ n . It is proved that for any given sequence of nonnegative integers ( i 2,…, i n ) satisfying 0⩽i k⩽( n k ) and any given nondegenerate probability distributions P 1,…, P n there exist random variables η 1,…, η n with respective distributions P 1,…, P n such that ( i 2,…, i n ) is their uncorrelation structure.