Suppose that $X_1,\cdots,X_n$ are iid standard Gaussian. $X_{(n)}$ is the maximum of $(X_1,\cdots,X_n)$, how can I find the asymptotic order of $VAR[X_{(n)}]$?
The density function of $X_{(n)}$ can be obtained as follows: $ P(X_{(n)}<t)=\prod P(X_i< t)=[\Phi(t)]^n, $ So the density is $f(t)=n\phi(t)[\Phi(t)]^{n-1}$.
But it's unclear to me about the approximation of the integral:
$\int_{-\infty}^\infty t^2n\phi(t)[\Phi(t)]^{n-1} dt$ which is the second moment.
I know that the first moment can be bounded by $$ E(X_{(n)})\le \sqrt{2\log n}-\frac{\log\log n+\log 4\pi - 2\gamma}{2\sqrt{2\log n}}. $$
It is known that the limiting distribution of $X_{(n)}$ of i.i.d. standard Gaussians is standard Gumbel. Meaning, the variance is $\frac {\pi ^2}{6}$. You just have to verify whether this is approx. or exact.