Limsup Random Fractals

Orey and Taylor (1974) introduced sets of “fast points” where Brownian increments are exceptionally large, F(λ) := {t ∈ [0, 1] : lim suph→0 |X(t+ h) −X(t)|/ √ 2h| log h|>λ}. They proved that for λ ∈ (0, 1], the Hausdorff dimension of F(λ) is 1 − λ2 a.s. We prove that for any analytic set E ⊂ [0, 1], the supremum of all λ’s for which E intersects F(λ) a.s. equals √ dimP E, where dimP 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 ∫ 1 0 |f ′|2 dt is lower than the packing dimension of E, there a.s. exists some t ∈ E so that f can be uniformly approximated in [0, 1] by normalized Brownian increments s 7→ [X(t+ sh)−X(t)]/√2h| log h|; such uniform approximation is a.s. impossible if the energy of f is higher than the packing dimension of E.