Laws of the iterated logarithm for α-time brownian motion

Abstract

We introduce a class of iterated processes called α-time Brownian motion for 0 < α < 2. We prove that there are universal constants cR, cL ∈ (0, ∞) such that lim supt → ∞ R*(t)/(t/log log t)1/2αlog log t = cR a.s. lim inft → ∞ supx ∈ R L*(x, t)/(t/log log t)1-1/2α = cL a.s. © 2006 Applied Probability Trust.