  We study generalizations of It\^{o}-Langevin dynamics consistent within
nonextensive thermostatistics. The corresponding stochastic differential
equations are shown to be connected with a wide class of nonlinear
Fokker-Planck equations describing correlated anomalous diffusion in fractals.
A generalized central limit theorem is proposed in order to demonstrate how
such equations emerge as a limit of correlated random variables. In doing so,
we connect microscopic and macroscopic descriptions of correlated anomalous
diffusion in a mathematically sound way and shed some light in explaining why
$q$-Gaussian distributions appear quite often in nature.
