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 Electronic Journal of Probability > Vol. 5 (2000) > Paper 4 open journal systems 


Limsup Random Fractals

Davar Khoshnevisan, University of Utah
Yuval Peres, University of California, Berkeley
Yimin Xiao, University of Utah


Abstract
Orey and Taylor (1974) introduced sets of ``fast points'' where Brownian increments are exceptionally large,
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 $sqrt{dimp E }$, where $dimp E$ is the {em 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 $E$.


Full text: PDF

Pages: 1-24

Published on: February 9, 2000
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Electronic Journal of Probability. ISSN: 1083-6489