Optimality

Copulas, information, dependence and decoupling 187
In what follows, F(x1, . . . , xn), xi∈ R, i = 1, . . . , n, stands for a function satisfying
the following conditions:
(a) F(x1, . . . , xn) = P(X1≤ x1, . . . , Xn≤ xn) for some r.v.’s X1, . . . , Xn on a
probability space (Ω,ℑ, P);
(b) the one-dimensional marginal cdf’s of F are F1, . . . , Fn;
(c) F is absolutely continuous with respect to dF(x1)···dFn(xn) in the sense
that there exists a Borel function G : R n → [0,∞) such that
F(x1, . . . , xn) =
� x1
−∞
� xn
··· G(t1, . . . , tn)dF1(t1)···dFn(tn).
−∞
dF
As usual, throughout the paper, we denote G in (c) by . In addi-
dF1···dFn
tion, F(xj1, . . . , xjk ), 1 ≤ j1 < ··· < jk ≤ n, k = 2, . . . , n, stands for the
k-dimensional marginal cdf of F(x1, . . . , xn). Also, in what follows, if not stated
otherwise, dF(xj1 ,...,xj ) k , 1≤j1