Individual behaviors of oriented walks

Given an infinite sequence t = (epsilon (k))(k) of -1 and +1, we consider t he oriented walk defined by S-n(t) = Sigma (n)(k=1) epsilon (1)epsilon (2). ..epsilon (k). The set of t's whose behaviors satisfy S-n(t) similar to bn( tau) is considered (b epsilon R and 0 < <tau> less than or equal to 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 re mains open. (C) 2000 Elsevier Science B.V. All rights reserved. MSG: 28A78; 42A55; 60G50.