  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 $\partial_t \rho=\partial^2_{xx}\rho^\nu$ ($\nu\neq1$).
