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 $E$.
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