Sets of random variables with a given uncorrelation structure

Abstract

Let xi (1),...,xi (n) be random variables having finite expectations. Denote i(k) := # {(j(1),...,j(k)): 1 less than or equal to j(1) < ... < j(k) less than or equal to n and E (l=1)pi (k) xi (fi) = (l=1)pi (k) E xi (h)}, k = 2,...,n. The finite sequence (i(2),...,i(n)) is called the uncorrelation structure of xi (1),...,xi (n). It is proved that for any given sequence of nonnegative integers (i(2),...,i(n)) satisfying 0 less than or equal to i(k) less than or equal to ((n)(k))and any given nondegenerate probability distributions P-1,...,P-n there exist random variables eta (1),...,eta (n) with respective distributions P-1,...,P-n such that (i(2),...,i(n)) is their uncorrelation structure. (C) 2001 Elsevier Science B.V. All rights reserved.