Lévy flight approximations for scaled transformations of random walks

Abstract

Complex systems under anomalous diffusive regime can be modelled by approximating sequences of random walks, S n = X 1 + X 2 + ⋯ + X n , where the i.i.d. random variables X j '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 distribution L α , σ , we present consistent estimates for the stability index α and for the scaling parameter σ . 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.