Optimality

Copulas, information, dependence and decoupling 189
Theorem 3.1 (Sklar [72]). If X1, . . . , Xn are random variables defined on a common
probability space, with the one-dimensional cdf’s FXk (xk) = P(Xk≤ xk) and
the joint cdf FX1,...,Xn(x1, . . . , xn) = P(X1≤ x1, . . . , Xn≤ xn), then there exists
an n-dimensional copula CX1,...,Xn(u1, . . . , un) such that FX1,...,Xn(x1, . . . , xn) =
CX1,...,Xn(FX1(x1), . . . , FXn(xn)) for all xk∈ R, k = 1, . . . , n.
The following theorems give analogues of the representations in the previous
section for copulas. Let V1, . . . , Vn denote independent r.v.’s uniformly distributed
on [0, 1].
Theorem 3.2. A function C : [0,1] n → [0, 1] is an absolutely continuous
n-dimensional copula if and only if there exist functions ˜gi1,...,ic : Rc→ R, 1≤
i1