Expectation of the maximum of standard Gaussians

Question

I'm strugging to find a neat solution to the following problem.

Let $X_1,\dots,X_N \sim N(0,1)$ and let $X = \max_i X_i$. Show that

$\mathbb{E}[X] \geq c \sqrt{\ln N}$

for some absolute constant $c$.

Combining the lower tail bound for standard Gaussians

$\mathbb{P} (X_i > a) \geq \frac{1}{\sqrt{2 \pi} a + 2} e^{-a^2/2}$

with the total expecation theorem $\mathbb{E}[X] = \mathbb{E}[X | X \geq a] \mathbb{P}(X \geq a) + \mathbb{E}[X | X < a]\mathbb{P}(X < a)$, I managed to prove the inequaility but it requeired a lot of tedious calculations.

Is there a simple and neat way to prove that lower bound?

I have seen a related post Expectation of the maximum of gaussian random variables, where the distribution of $X$ is derived, however I was not able to bound from below the expectation integral.

Many thanks in advance.

Answer 1

The way you said you proved the inequality is probably the only way to do it. It does not seem to require a lot of tedious calculations.

If $g$ is a standard Gaussian, then $$\hbox{Prob}\{g>t\}\geq (\frac{1}{t}-\frac{1}{t^3})e^{-t^2/2}$$ (See Feller, Vol. $1$, $3$rd edition, p. $175$) Hence there exists a positive $\beta>0$ (smaller than $1$) such that $$\hbox{Prob}\{g>\beta(\log n)^{1/2}\}\geq\frac{1}{n}$$ Now if $g_1,\dots,g_n$ are independent standard Gaussians, then using the previous inequality, we get $$\hbox{Prob}\{\max g_i \leq\beta(\log n)^{1/2}\}<(1-\frac{1}{n})^n\approx \frac{1}{e}$$ Hence $$\mathbb{E}(\max g_i)>\beta(\log n)^{1/2}\hbox{Prob}\{\max g_i>\beta(\log n)^{1/2}\}>(1-\frac{1}{e})\beta(\log n)^{1/2}$$