  We briefly review Boltzmann-Gibbs and nonextensive statistical mechanics as
well as their connections with Fokker-Planck equations and with existing
central limit theorems. We then provide some hints that might pave the road to
the proof of a new central limit theorem, which would play a fundamental role
in the foundations and ubiquity of nonextensive statistical mechanics. The
basic novelty introduced within this conjectural theorem is the {\it
generalization of the hypothesis of independence} of the $N$ random variables
being summed. In addition to this, we also advance some nonlinear dynamical
(possibly exact) relations which generalize the concepts of Lyapunov exponents,
entropy production per unit time, and their interconnection as first proved by
Pesin for chaotic systems.
