The stochastic properties of variables whose addition leads to $q$-Gaussian distributions $G_q(x)=[1+(q-1)x^2]_+^{ 1/(1-q)}$ (with $q\in\mathbb{ R}$ and where $[f(x)]_+=max\{ f(x),0\}$) as limit law for a large number of terms are investigated. These 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 have important consequences for a deeper understanding of the domain of applicability of such generalization. Basically, it is shown that the random walk sequences, that are relevant to this problem, possess a simple additive-multiplicative structure commonly found in many contexts, thus justifying the ubiquity of those distributions. Furthermore, a connection is established between such sequences and the nonlinear diffusion equation $\partial_t \rho=\partial^2_{ xx}\rho^\nu$ ($\nu\neq1$).
Categories: cond-mat.stat-mech