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

TL;DR

This paper investigates the limit distributions of scale-invariant correlated random variables, demonstrating that q-Gaussians emerge naturally under certain invariance conditions and linking them to the Boltzmann-Gibbs entropy.

Contribution

It introduces a generalized stochastic model that produces q-Gaussians for all q values, including q>1, using the Laplace-de Finetti theorem and scale-invariance principles.

Findings

q-Gaussians arise as limit distributions for scale-invariant correlated variables

Strict scale invariance with q-Gaussianity implies Boltzmann-Gibbs entropy

The model encompasses all q values, extending previous results

Abstract

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.