Limsup Random Fractals

Abstract

Orey and Taylor (1974) introduced sets of ``fast points'' where Brownian increments are exceptionally large, ${\rm F}(\lambda):=\{ t\in[0,1]: \limsup_{h\to 0}{ | X(t+h)-X(t)| / \sqrt{ 2h|\log h|}} \ge \lambda\}$. They proved that for $\lambda \in (0,1]$, the Hausdorff dimension of ${\rm F}(\lambda)$ 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 ${\rm F}(\lambda)$ a.s. equals $\sqrt{\text{dim}_p E }$, where $\text{dim}_p E$ 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 Deheuvels and Mason (1994). In particular, we prove that for any absolutely continuous function $f$ such that $f(0)=0$ and the energy $\int_0^1 |f'|^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 $E$.