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

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