On the Hausdorff dimension of the set generated by exceptional oscillations of a wiener process

P. Deheuvels, M. A. Lifshits

Research output

8 Citations (Scopus)

Abstract

The rescaled h-increments Yt,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 C0[0,1] of all continuous functions g on [0,1] with g(0) = 0. We endow C0[0,1] with the topology defined by a norm ∥·∥ν chosen within a general class C for which the limit law limh↓0{sup0≤t≤1 ∥Yt,hν} < ∞ holds with probability 1. We show that, for each f ∈ C0[0,1] with ∫0{d/du f(u)}2du≤ 1, the set ℒν(f)={t ∈ [0,1]: lim infh↓0∥Yt,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)}2du.

Original languageEnglish
Pages (from-to)75-110
Number of pages36
JournalStudia Scientiarum Mathematicarum Hungarica
Volume33
Issue number1-3
Publication statusPublished - 1 Dec 1997

Scopus subject areas

  • Mathematics(all)

Fingerprint Dive into the research topics of 'On the Hausdorff dimension of the set generated by exceptional oscillations of a wiener process'. Together they form a unique fingerprint.

Cite this