## Abstract

The rescaled h-increments Y_{t,h}(u) = (2h log(1/h))-^{1/2}{W(t + hu) - W(t)}, for u ∈ [0,1], of a Wiener process {W(t) : t ≥ 0}, are considered as elements of the space C_{0}[0,1] of all continuous functions g on [0,1] with g(0) = 0. We endow C_{0}[0,1] with the topology defined by a norm ∥·∥_{ν} chosen within a general class C for which the limit law limh↓0{sup_{0}≤t≤1 ∥Y_{t,h}∥_{ν}} < ∞ holds with probability 1. We show that, for each f ∈ C_{0}[0,1] with ∫_{0}{d/du f(u)}^{2}du≤ 1, the set ℒ_{ν}(f)={t ∈ [0,1]: lim inf_{h↓0}∥Y_{t,h}-f∥_{ν}=0} contains, with probability 1 for each ν ∈ C, a subset ℒ(f), independent of ∥·∥ C and with Hausdorff dimension equal to dim(ℒ(f)) = 1 - ∫_{0} ^{1} {d/du f (u)}^{2}du.

Original language | English |
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Pages (from-to) | 75-110 |

Number of pages | 36 |

Journal | Studia Scientiarum Mathematicarum Hungarica |

Volume | 33 |

Issue number | 1-3 |

Publication status | Published - 1 Dec 1997 |

## Scopus subject areas

- Mathematics(all)