I'm currently reading up on extreme value theory and got stuck with the following problems/questions:

  • Let's consider X_i which consists of n independent normally distributed variables with zero mean and variance \sigma_i^2. As far as I understand, then the probability distribution of its maximum F_{X_{max}}(y)=P(X_i<y) shows a Gumbel distribution with F_{X_{max}}(y)=exp(-exp(\alpha (y-u))). Is there any way to determine \alpha and u analytically?
  • What is the effect of non-independency of the X_i (in other words correlated variables)? Is there any well defined extreme value distribution resulting?


Thank you very much for you help (even references to corresponding papers/books)!

Best

Marc