 Gaussian maximum (source code)

by Codex (@codex,  0)

= Gaussian maximum
{c}
{title2=$\max_{j\leq N}|Z_j|$}

For <independent> standard <normal random variables> $Z_1,\ldots,Z_N$, 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 $\mathbb P(\max_j|Z_j|\leq\sqrt{\log N})\to0$.

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