Extreme Value Distribution of dependent RV Let $X_1, \dots, X_n$ be random variables with known identical distribution and covariance matrix $C \in \mathbb{R}^{n\times n}$. How can I model / calculate the extreme value distribution:
$$ f(w) = P( X_1 \leq w, \dots,  X_n \leq w) $$
I'm not familiar with this topic and welcome any help / hints to learn. Is there any information missing to specify this problem? In my application $n \approx 10000$. $C$ is a Toeplitz matrix. 
 A: There is a result known as Sklar's theorem which is useful to consider here.  It states that you can decompose the joint distribution function $G$ of a random vector $(X_1, X_2, \ldots , X_n)$ as
\begin{align}
G(s) &=P(X_1 \leq s_1 \cap X_2 \leq s_2 \cap \ldots \cap X_n \leq s_n) \\
&= C[F_1(s_1), F_2(s_2), \ldots , F_n(s_n)]
\end{align}
where $F_i(s) = P(X_i \leq s)$ is the marginal distribution function of $X_i$ and $C$ is the joint distribution function for a particular multivariate random vector $(U_1, U_2, \ldots , U_n)$ where each $U_i$ marginally follows of a uniform distribution over $(0, 1)$.  In general the $U_i$ are not independent, and their dependence structure is captured by $C$.
So if you are willing to assume something about this copula as well as the common marginal distribution $F$ then you could determine the distribution function of the maximum
\begin{align}
P(\max(X_1, \ldots , X_n) \leq s) &= P(X_1 \leq s \cap X_2 \leq s \cap \ldots \cap X_n \leq s) \\
&= C[F(s), F(s) , \ldots , F(s)] .
\end{align}
Do you know or at least have a way of estimating these components?
