I'm looking for any result about asymptotic behaviour of a maxima of non identically independent rv's.
Actually, i want to get an order, when $n$ goes to infinity, of $M_n = \max(X_1,..,X_n)$ where $(X_i)$ are independent but $\textbf{non identically distributed}$. In other words,
In my example, $X_n \sim \chi(\lambda n)$, where $\lambda n$ stay bounded. For instance, we may choose $\lambda = \lambda(n) = \frac{1}{n+1}$.
I have been searching all over the litterature (Galambos, Resnick, Leadbetter) of extreme value theory, but in vain. It just came up that for i.i.d $\chi$-distribution, the order is about $\sqrt{\log(n)}$, modulo some multiplicative constant.
Any help will be thankfully welcome :)