TY - JOUR

T1 - On the Hausdorff dimension of the set generated by exceptional oscillations of a wiener process

AU - Deheuvels, P.

AU - Lifshits, M. A.

PY - 1997/12/1

Y1 - 1997/12/1

N2 - 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,h∥ν} < ∞ 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.

AB - 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,h∥ν} < ∞ 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.

KW - Fractals

KW - Hausdorff dimension

KW - Law of the iterated logarithm

KW - Modulus of continuity

KW - Strong laws

KW - Wiener process

UR - http://www.scopus.com/inward/record.url?scp=0002815170&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0002815170

VL - 33

SP - 75

EP - 110

JO - Studia Scientiarum Mathematicarum Hungarica

JF - Studia Scientiarum Mathematicarum Hungarica

SN - 0081-6906

IS - 1-3

ER -