Individual behaviors of oriented walks

Given an infinite sequence t=(εk)k of −1 and +1, we consider the oriented walk defined by Sn(t)=∑k=1nε1ε2…εk. The set of tʹs whose behaviors satisfy Sn(t)∼bnτ is considered (b∈R and 0<τ⩽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.