  Extremization of the Boltzmann-Gibbs (BG) entropy under appropriate norm and
width constraints yields the Gaussian distribution. Also, the basic solutions
of the standard Fokker-Planck (FP) equation (related to the Langevin equation
with additive noise), as well as the Central Limit Theorem attractors, are
Gaussians. The simplest stochastic model with such features is N to infinity
independent binary random variables, as first proved by de Moivre and Laplace.
What happens for strongly correlated random variables? Such correlations are
often present in physical situations as e.g. systems with long range
interactions or memory. Frequently q-Gaussians become observed. This is
typically so if the Langevin equation includes multiplicative noise, or the FP
equation to be nonlinear. Scale-invariance, i.e. exchangeable binary stochastic
processes, allow a systematical analysis of the relation between correlations
and non-Gaussian distributions. In particular, a generalized stochastic model
yielding q-Gaussians for all q (including q>1) was missing. This is achieved
here by using the Laplace-de Finetti representation theorem, which embodies
strict scale-invariance of interchangeable random variables. We demonstrate
that strict scale invariance together with q-Gaussianity mandates the
associated extensive entropy to be BG.
