I need the expected value of the empirical copula.
I have for a fix $\mathbf{u}=(u_1,\dots,u_d) \in [0,1]^d$
$\mathbb{E}[C^n(\mathbf{u})] = \frac{1}{n} \sum_{i=1}^n \mathbb{E}[\mathbf{1}(\tilde{U}_1^i \le u_1,\dots,\tilde{U}_d^i \le u_d)] $
Then
$\mathbb{E}[\mathbf{1}(\tilde{U}_1^i \le u_1,\dots,\tilde{U}_d^i \le u_d)] = \mathbb{P}(\tilde{U}_1^i \le u_1,\dots,\tilde{U}_d^i \le u_d) = \mathbb{P}(F^n_1(X^i_1) \le u_1,\dots,F^n_d(X^i_d) \le u_d)$
Now I stuck here and can't see how to get a good result.
Any idea?
With best regards
Fluffy