= Supremum norm risk of a Haar projection estimator
{title2=$E\|\widehat\Pi_{V_J}f-\Pi_{V_J}f\|_\infty\leq\sqrt{2^J(2J+2)\log2/n}$}
The cellwise error of the <Haar projection estimator in Gaussian white noise> is $\sqrt{2^J/n}$ times a standard normal variable. Its <supremum norm> is therefore exactly that factor times the <Gaussian maximum> over $2^J$ cells. The <Gaussian maximum bound without independence> gives expected error at most $\sqrt{2^J(2J+2)\log2/n}$. This concerns the stochastic error about the projection; approximation bias must be added when estimating the full drift.
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