Lévy flight approximations for scaled transformations of random walks

Complex systems under anomalous diffusive regime can be modelled by approximating sequences of random walks, Sn =X1 + X2 + · · · + Xn, where the i.i.d. random variables Xj ’s have fat-tailed distribution. Such random walks are referred by physicists as Lévy flights or motions and have been used to model financial data. For better adjustment to real-world data several modified Lévy flights have been proposed: truncated, gradually truncated or exponentially damped Lévy flights. On the other hand, scaled transformations of random walks possess a Lévy stable limiting distribution. In this work, under the assumption that the analyzed data belongs to the domain of attraction of a symmetric Lévy stable distributionL , , we present consistent estimates for the stability index and for the scaling parameter . Variations of the model that allow distinct left and right tail behavior will be explored. Illustrations for returns of exchange rates of several countries are also included. © 2007 Elsevier B.V. All rights reserved.