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+⋯+XnSn=X1+X2+⋯+Xn, where the i.i.d. random variables XjXj'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 Lα,σLα,σ, we present consistent estimates for the stability index αα and for the scaling parameter σσ. 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.