# Why do copulas need the i.i.d assumption for marginal distribution?

Does anyone know if are there some assumptions for Copula method? I heard from someone that the data should be i.i.d (independent and identically distributed). Let's say, if I want to capture the dependence structure between two variables. I have to use marginal distribution to transform data to rank space, then I can fit a theoretical Copula function. Is there an assumption for the marginals that the data should be i.i.d? If so, why? And does one apply Copula method if the time series are not i.i.d?

The reason is that you're looking for a wrong joint. Consider this, you're trying to recover the joint distribution $$F(X,Y)$$ with copulas, where $$X,Y$$ are random variables. However, in case of first order serial correlation in observations (autocorrelation) the joint distribution is $$F(X_t,X_{-1},Y_t,Y_{t-1})$$ not the one you're trying to recover.
Copulas are based on Sklar's theorem and this theorem does not require independency. Besides, there exists an independence copula such that: $$F(x_1, \ldots , x_d ) = \Pr(X_1 ≤ x_1, \ldots , X_d ≤ x_d )$$ $$= \Pr(X_1 ≤ x_1)\cdots\Pr(X_d ≤ x_d ) = F_1(x_1)\cdots F_d(x_d)$$ and $$F(x_1, \ldots , x_d )$$ $$= C(F_1(x_1), \ldots , F_d(x_d))$$