Lvy 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 S n = X 1 + X 2 + ? + X n , where the i.i.d. random variables Xj X j 's have fat-tailed distribution. Such random walks are referred by physicists as Lvy flights or motions and have been used to model financial data. For better adjustment to real-world data several modified Lvy flights have been proposed: truncated, gradually truncated or exponentially damped Lvy flights. On the other hand, scaled transformations of random walks possess a Lvy stable limiting distribution. In this work, under the assumption that the analyzed data belongs to the domain of attraction of a symmetric Lvy stable distribution L?,? L ? , ? , 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.
lvy flights stability index hill's estimate foreign exchange rates