Computational Statistics & Data Analysis 51 (2007) 6343 – 6354 www.elsevier.com/locate/csda Lévy flight approximations for scaled transformations of random walks  Chang C.Y. Dorea a , ∗ , Cira E. Guevara Otiniano a , Raul Matsushita b , Pushpa N. Rathie b a Depto. de Matemática, Universidade de Brasilia, 70910-900 Brasilia-DF, Brazil b Depto. de Estatística, Universidade de Brasilia, 70910-900 Brasilia-DF, Brazil Received 26 January 2006; received in revised form 19 October 2006; accepted 24 January 2007 Available online 4 February 2007 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 , , 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. Keywords: Lévy flights; Stability index; Hill’s estimate; Foreign exchange rates 1. Introduction Consider the random walk S(n) = X 1 + X 2 +···+ X n , where {X k } k  1 is a sequence of independent and identically distributed (i.i.d.) random variables with zero-mean but not necessarily finite variance. The asymptotic behavior of the normalized partial sums may be non-Gaussian and 1 n E{S(n)}→∞, which indicates an anomalous diffusive regime. Such random walks are commonly referred to by physicists as Lévy flights or Lévy motions. On the other hand, a stable distribution can be considered as a limiting (in law) probability distribution of the normalized sums of i.i.d. random variables. Moreover, if X 1 ,X 2 ,... are independent copies of X d = L , ( d =: equality in law; L , : symmetric Lévy stable distribution with stability index  and scaling  Research partially supported by CNPq, FAPDF/PRONEX, CAPES/PROCAD and FINATEC/UnB. ∗ Corresponding author. Tel.: +55 6132733356; fax: +55 6132732737. E-mail address: cdorea@mat.unb.br (C.C.Y. Dorea). 0167-9473/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2007.01.019