Iterated law of iterated logarithm

Abstract

Suppose [epsilon] [is a member of the set] [0, 1) and let theta [subscipt epsilon] (t) = (1 − [epsilon]) [square root of] (2tln [subscript] 2 t). Let L [to the power of epsilon] [subscript] t denote the amount of local time spent by Brownian motion on the curve [theta subscript epsilon] (s) before time t. If [epsilon] > 0 then lim sup [subscript] t [to infinity] L [to the power of epsilon] [subscript] t / [square root of] (2tln [subscript] 2 t) = 2 [epsilon] + o ([epsilon]). For [epsilon] = 0, a non-trivial limsup result is obtained when the normalizing function [square root of] (2tln [subscript] 2 t) is replaced by g(t) = [square root of] (t / ln [subscript] 2 t) ln [subscript] 3 t.