Non-extensive random walks

Abstract

Stochastic variables whose addition leads to q-Gaussian distributions Gq(x)∝[1+(q-1)βx2]+1/(1-q) (with β>0, 1≤q<3 and where [f(x)]+=max{f(x),0}) as limit law for a large number of terms are investigated. Random walk sequences related to this problem possess a simple additive-multiplicative structure commonly found in several contexts, thus justifying the ubiquity of those distributions. A characterization of the statistical properties of the random walk step lengths is performed. Moreover, a connection with non-linear stochastic processes is exhibited. q-Gaussian distributions have special relevance within the framework of non-extensive statistical mechanics, a generalization of the standard Boltzmann-Gibbs formalism, introduced by Tsallis over one decade ago. Therefore, the present findings may give insights on the domain of applicability of such generalization. © 2005 Elsevier B.V. All rights reserved.