Let $(N_k)_{k\geqslant1}$ be a sequence of independent standard normal stochastic variables, and let $X_n=|\max(N_1,\ldots,N_n)-\min(N_1,\ldots,N_n)|$. I'm wondering whether $\mathbb E(X_n)$ diverges. Also, if we define $X_{n,\alpha}$ such that $\mathcal P(X_n>X_{n,\alpha})=\alpha$, for which $\alpha\in]0,1[$ does $X_{n,\alpha}$ diverge?
Motivation: this question came up when studying the asymptotic behaviour of the Studentized Range Distribution when constructing confidence intervals.