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

TL;DR

This paper reviews classical limit theorems and explores the behavior of q-Gaussian distributions, derived from nonextensive entropy, under convolution and the q-product, revealing their attractors and correlation structures.

Contribution

It introduces the application of classical limit theorems to q-Gaussian distributions and discusses the role of the q-product in nonextensive statistical mechanics.

Findings

q-Gaussian has Gaussian or Levy stable attractors depending on q

Finite variance for q < 5/3, infinite for q > 5/3

The q-product is central to correlated variable analysis

Abstract

In this article we review the standard versions of the Central and of the Levy-Gnedenko Limit Theorems, and illustrate their application to the convolution of independent random variables associated with the distribution known as q-Gaussian. This distribution emerges upon extremisation of the nonadditive entropy, basis of nonextensive statistical mechanics. It has a finite variance for q < 5/3, and an infinite one for q > 5/3. We exhibit that, in the case of (standard) independence, the q-Gaussian has either the Gaussian (if q < 5/3) or the a-stable Levy distributions (if q > 5/3) as its attractor in probability space. Moreover, we review a generalisation of the product, the q-product, which plays a central role in the approach of the specially correlated variables emerging within the nonextensive theory.