Gaussian maximum (${\max }_{j\le N}|Z_j|$)

For independent standard normal random variables $Z_1,\dots ,Z_N$, the maximum of $|Z_j|$ has scale $\sqrt{\log N}$. A union bound and the Gaussian tail bound give the upper scale. Integrating the normal density over $[r,r+1/r]$ gives a lower tail bound $c{r}^{-1}{e}^{-r^2/2}$, implying $\mathbb{P}({\max }_j|Z_j|\le \sqrt{\log N})\to 0$.

For $N$ variables with the standard normal distribution, the expected maximum of their absolute values is at most $\sqrt{2\log (2N)}$, even when they are dependent. Bound the exponential of the maximum by the sum of the $2N$ signed exponentials, use the normal moment-generating function, then apply Jensen inequality and minimize over the exponential parameter. The resulting logarithmic growth is useful in simultaneous estimation.