Individual behaviors of oriented walks

Given an infinite sequence t=([var epsilon]k)k of -1 and +1, we consider the oriented walk defined by Sn(t)=[summation operator]k=1n[var epsilon]1[var epsilon]2...[var epsilon]k. The set of t's whose behaviors satisfy Sn(t)~bn[tau] is considered ( and 0<[tau][less-than-or-equals, slant]1 being fixed) and its Hausdorff dimension is calculated. A two-dimensional model is also studied. A three-dimensional model is described, but the problem remains open.