Limsup random fractals. (English) Zbl 0949.60025

Khoshnevisan, Davar; Peres, Yuval; Xiao, Yimin

Electron. J. Probab. 5, Paper No. 4, 24 p. (2000).

Summary: S. Orey and S. J. Taylor [Proc. Lond. Math. Soc., III. Ser. 28, 174-192 (1974; Zbl 0292.60128)] introduced sets of “fast points” where Brownian increments are exceptionally large, \(F(\lambda):=\{t\in[0,1]:\limsup_{h\to 0}|X(t+h)-X(t)|/{\sqrt{2h|\log h|}}\geqslant\lambda\}\). They proved that for \(\lambda \in (0,1]\), the Hausdorff dimension of \(F(\lambda)\) is \(1-\lambda^2\) a.s. We prove that for any analytic set \(E \subset [0,1]\), the supremum of the \(\lambda\) such that \(E\) intersects \(F(\lambda)\) a.s. equals \(\sqrt{\dim_{\text{P}}E }\), where \(\dim_{\text{P}} E\) is the packing dimension of \(E\). We derive this from a general result that applies to many other random fractals defined by limsup operations. This result also yields extensions of certain “fractal functional limit laws” due to P. Deheuvels and D. M. Mason [in: Probability in Banach spaces, 9. Prog. Probab. 35, 73-89 (1994; Zbl 0809.60042)]. In particular, we prove that for any absolutely continuous function \(f\) such that \(f(0)=0\) and the energy \(\int_0^1 |f^\prime|^2 dt \) is lower than the packing dimension of \(E\), there a.s.exists some \(t \in E\) so that \(f\) can be uniformly approximated in \([0,1]\) by normalized Brownian increments \(s \mapsto [X(t+sh)-X(t)] / \sqrt{ 2h|\log h|}\); such uniform approximation is a.s. impossible if the energy of \(f\) is higher than the packing dimension of \(E\).

MSC:

60D05	Geometric probability and stochastic geometry
60G17	Sample path properties
28A80	Fractals

Keywords:

packing dimension; Hausdorff dimension; Brownian motion; fast point

Citations:

Zbl 0292.60128; Zbl 0809.60042