Nonextensive statistical mechanics and central limit theorems II - Convolution of q-independent random variables

TL;DR

This paper reviews recent advances in nonextensive statistical mechanics, focusing on the central limit theorem for q-independent variables, highlighting the emergence of q-Gaussians and q-stable distributions as attractors.

Contribution

It introduces a generalization of the central limit theorem for q-independent variables, connecting q-Gaussians and q-stable distributions as attractors in probability space.

Findings

q-Gaussians emerge as attractors for q ≥ 1

A q-generalization of stable Levy distributions is discussed

Identification of a triplet of entropic indices relating attractors, correlations, and scaling

Abstract

In this article we review recent generalisations of the central limit theorem for the sum of specially correlated (or q-independent) variables, focusing on q greater or equal than 1. Specifically, this kind of correlation turns the probability density function known as q-Gaussian, which emerges upon maximisation of the entropy Sq, into an attractor in probability space. Moreover, we also discuss a q-generalisation of a-stable Levy distributions which can as well be stable for this special kind of correlation.Within this context, we verify the emergence of a triplet of entropic indices which relate the form of the attractor, the correlation, and the scaling rate, similar to the q-triplet that connects the entropic indices characterising the sensitivity to initial conditions, the stationary state, and relaxation to the stationary state in anomalous systems.