In this Note, we establish a de Acosta (1985) type strong law for a family of Hölder norms. More precisely, we obtain, for α∈(0,1/2), the exact rate of convergence, as h↓0, of
\[
T_{\mathrm{\alpha ,f}}\mathrm{(h):=}\underset{\mathrm{0⩽t⩽1-h}}{\mathrm{inf}}\mathrm{\Vert (2h}\mathrm{log}\mathrm{(1/h))}^{\mathrm{-1/2}}\mathrm{(W(t+h·)-W(t))-f\Vert }_{\alpha }
\] when $ \mathrm{f\in }\mathcal{S}$
... [Show full abstract] satisfies $ \int _{0}^{1}\{\frac{\mathrm{d}}{\mathrm{d}u}\mathrm{f(u)\}}^{2}\text{\hspace{0.17em}}\mathrm{d}\mathrm{u