Suppose that I have a sequence of $n$ i.i.d. standard Gaussian random variables $X_1,\ldots,X_n$ where $X_i\sim\mathcal{N}(0,1)$. Denote the maximum of this sequence by $M_n=\max(X_1,\ldots,X_n)$.
I am wondering about the dependence between the maximum $M_n$ and some random variable in that sequence $X_i$ as $n\rightarrow\infty$. My intuition tells me that $M_n$ should be independent of any given $X_i$ as the sequence gets larger, since then an individual member of the sequence, $X_i$, doesn't "contribute" much to the maximum. However, I am not sure whether this is in fact true. Does anyone know of the existence of a formal proof?
If it turns out that maximum of this sequence is asymptotically independent of a particular random variable in the sequence, this would be helpful for the computation of a limit of expected value $\lim_{n\rightarrow\infty}E[f(X_i)g(M_n)]$ where expectations $E[f(X_i)]$ and $E[g(M_n)]$ can be easily computed.