Supremum norm risk of a Haar projection estimator ($E\Vert {\hat{\Pi }}_{V_J}f-{\Pi }_{V_J}f{\Vert }_{\infty }\le \sqrt{2^J(2J+2)\log 2/n}$)

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 $\sqrt{2^J(2J+2)\log 2/n}$. This concerns the stochastic error about the projection; approximation bias must be added when estimating the full drift.