Optimality

Copulas, information, dependence and decoupling 201
R.v’s with the joint distribution function (8.6) are independent. Let now X1, . . . , Xn
be independent r.v.’s with one-dimensional distribution functions Fi(xi), i =
1, . . . , n. Then their joint distribution function has form (8.6). This and the uniqueness
of the functions gi1,...,ic given by Theorem 2.1 completes the proof of the
theorem.
Proof of Theorems 5.2–5.8. Below, we give proofs of Theorems 5.7 and 5.8.
The rest of the theorems can be proven in a similar way. Let X1, . . . , Xn be r.v.’s
with the joint distribution function satisfying representation (2.2) with functions
gi1,...,ic such that Eξ αi 1
i1 ···ξ αic
ic gi1,...,ic(ξi1, . . . , ξic) = 0, 1≤i1