A reference for a Gaussian inequality ($\mathbb{E} \max_i X_i$)
I eventally found these two references:
- from [1]: the expected value of the maximum of $N$ independent standard Gaussians: Theorem 2.5 and Exercise 2.17, p. 49; for a concentration result, combined with the variance (which is $O(1)$). Exercise 3.24 (or Theorem 5.8 for directly a concentration inequality).
- from [2], Theorem 3.12
[1] Concentration Inequalities: A Nonasymptotic Theory of Independence By Stéphane Boucheron, Gábor Lugosi, Pascal Massart (2013)
[2] Concentration Inequalities and Model Selection, by Pascal Massart (2003)
You have an explicit asymptotic result concerning the limit distribution in this post and the associated references.