= Gaussian maximum bound without independence
{c}
{title2=$E\max_{i\leq N}|g_i|\leq\sqrt{2\log(2N)}$}
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.
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