Scale-Invariant Correlated Probabilistic Model Yields q-Gaussians in the Thermodynamic Limit

Extremization of the Boltzmann-Gibbs (BG) entropy SBG = −k R dx p(x) ln p(x) under appropriate norm and width constraints yields the Gaussian distribution pG(x) ∝ e− x2 . 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 → ∞ independent binary random variables, as first proved by de Moivre and Laplace. These well known results mathematically ground BG statistical mechanics. What happens for strongly correlated random variables? Such correlations are often present in physical situations, which are often characterized by q-Gaussians, pq(x) ∝ [1 − (1 − q) x2]1/(1−q) [p1(x) = pG(x)]. It is typically so if we allow the Langevin equation to include multiplicative noise, or the FP equation to be nonlinear. The ubiquitous property of scale-invariance enables a systematical analysis of the relation between correlations and non-Gaussian distributions. Nevertheless, a generalized stochastic model yielding q-Gaussians (q 6= 1) was missing. This is achieved here by using the Laplace-de Finetti representation, which embodies strict scale-invariance of interchangeable N random variables. We also demonstrate that strict scale invariance together with q-Gaussianity mandates the associated extensive entropy to be that of BG.