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On the functional form of Lévy’s modulus of continuity for Brownian motion. (English) Zbl 0548.60034

Let \(\{\) W(s): \(s\geq 0\}\) be a Brownian motion taking values in a Banach space B and for \(h>0\), \(\Delta (s,h)=W(s+(\cdot)h)-W((s)\), \(l(h)=(2h \log (h^{-1}))^{{1\over2}}\). For a fixed closed interval I in \({\mathbb{R}}^+\), let \({\mathcal F}(h)=\{\Delta (s,h):\) \(s\in I\}\). We prove that \(\lim_{h\downarrow 0}{\mathcal F}(h)/l(h)={\mathcal K}^ a.\)s. in the Hausdorff metric on the non-empty closed bounded subsets of C([0,1],B), where \({\mathcal K}\) is the unit ball of the reproducing kernel Hilbert space H associated to W. Furthermore, for \(f\in {\mathcal K}\) and \(\| f\|_ H<1\) we obtain the exact rate at which the random set \({\mathcal F}(h)/l(h)\) approaches f.

MSC:

60F17 Functional limit theorems; invariance principles
60J65 Brownian motion
60G17 Sample path properties
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