Let $X(n)$ be the largest observation in a random sample of size $n$ from a standard normal population. In this paper we investigate the limiting behavior of the distribution and the moments of $X(n)$ for large $n$. The results of this paper provide rates of convergence of the distribution and the moments of $X(n)$.
"Asymptotic Distribution and Moments of Normal Extremes." Ann. Probab. 9 (1) 150 - 153, February, 1981. https://doi.org/10.1214/aop/1176994515