Limit distributions of scale-invariant probabilistic models of correlated random variables with the q-Gaussian as an explicit example

Abstract

Extremization of the Boltzmann-Gibbs (BG) entropy $S_{BG}=-k\int dx\,p(x) \ln p(x)$ under appropriate norm and width constraints yields the Gaussian distribution p_G(x) ∝e^-βx. 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. 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, p_q(x) ∝[1-(1-q)βx²]^1/(1-q) [p₁(x)=p_G(x)] become observed. This is typically so if the Langevin equation includes multiplicative noise, or the FP equation to be nonlinear. Scale-invariance, e.g. 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 ≠ 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.