Iterated law of iterated logarithm

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