# Why does countermonotonicity induce lower Fréchet–Hoeffding copula bounds?

I am learning Copula that describes dependancy between random variables.

The upper bound of Fréchet–Hoeffding is quite easy to understand. However, I do not understand the intituivie behind the fact that the countermonotonic random varibles, not independent ones, induce the lower bound.

In this document (at the end of page 6), the author said that

Certainly, independence is quite contrary to this, and for two independent rvs the copula equals $C(u_1, u_2) = u_1 \cdot u_2$. However, independence just serves as an intermediate step on the way to the contrary extreme of comonotonicity, namely the case of countermonotonic rvs. In terms of uniform rvs this case is obtained for $U_2 = 1 − U_1$.

Could anyone please explain how independence is only an intermediate step?

• Welcome to MSE. Please use MathJax. – José Carlos Santos Apr 29 '18 at 10:36
• @JoséCarlosSantos done. – Alma Cantu Apr 29 '18 at 10:42