TITLE: On $Z = \max \left(X_1, X_2, \dots, X_N \right)$ where $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$
QUESTION [6 upvotes]: How does the maximum of $N$ normal random variables behave? If they are i.i.d., then on the right tail ($x \rightarrow \infty$) it behaves as if it is distributed with Gumbel. But any results for the generic case? 
That is, how does $Z = \max \left(X_1, X_2, \dots, X_N \right)$ behave where $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$, or with shared variance $X_i \sim \mathcal{N}(\mu_i, \sigma^2)$. 
I assume the shared variance & different mean case should behave similar to the i.i.d. case, but are there any known results?

REPLY [3 votes]: Ross 2003 gives an upper bound for the more general case, where $X_i$ can be dependent:
$$\mathbb{E} \left[ \max_i X_i \right] \le c + \sum_{i=1}^N \left(\sigma^2_i + f_i(c) + (\mu_i - c) Pr\left\{X_i > c \right\} \right)$$
for all values of $c$, where $f_i(c)$ is the pdf of $X_i$.
The tightest bound can be obtained by using a $c$ that satisfies
$$\sum_{i=1}^N Pr\left\{X_i > c \right\} = 1 \: . $$