Hausdorff-Besicovitch measure of fractal functional limit laws induced by Wiener process in Hölder norms. (Q2490803)

For a standard linear Brownian motion~\(W\) the authors find the precise rate of convergence of the uncountable family of rescaled increments \[ \big( (2h\log(1/h))^{-1/2} (W(t+hs)-W(t)) \colon 0\leq s \leq 1\big) , \quad \text{ for } 0\leq t\leq1-h, \] to elements of the Strassen set with respect to the \(\alpha\)-Hölder norm, for \(0<\alpha<2\). In a further result they find the Hausdorff dimension of the set of times~\(t\) where the increments \(((2h\log(1/h))^{-1/2} (W(t+hs)-W(t)) \colon 0\leq s \leq 1)\) approximate a given~\(f\) with the optimal rate. This result is in the spirit of \textit{P. Deheuvels} and \textit{D. M. Mason} [Stud. Sci. Math. Hung. 34, No.~1--3, 89--106 (1998; Zbl 0916.60037)] and more recent unifying results of \textit{D. Khoshnevisan, Y. Peres} and \textit{Y. Xiao} [Electron. J. Probab. 5, Paper No.~4 (2000; Zbl 0949.60025)].