On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics

TLDR: The standard central limit theorem plays a fundamental role in Boltzmann-Gibbs statistical mechanics and has been successfully applied to a considerable amount of physically interesting complex phenomena as mentioned in this paper.

Abstract: The standard central limit theorem plays a fundamental role in Boltzmann-Gibbs statistical mechanics. This important physical theory has been generalized [1] in 1988 by using the entropy \(S_{q} = \frac{1-\sum_{i} p^{q}_{i}}{q-1}\)\(({\rm with}\,q\,\in {{{\mathcal{R}}}})\) instead of its particular BG case \(S_{1} = S_{BG} = - \sum_{i} p_{i}\,{\rm ln}\,p_{i}\). The theory which emerges is usually referred to as nonextensive statistical mechanics and recovers the standard theory for q = 1. During the last two decades, this q-generalized statistical mechanics has been successfully applied to a considerable amount of physically interesting complex phenomena. A conjecture[2] and numerical indications available in the literature have been, for a few years, suggesting the possibility of q-versions of the standard central limit theorem by allowing the random variables that are being summed to be strongly correlated in some special manner, the case q= 1 corresponding to standard probabilistic independence. This is what we prove in the present paper for \(1{\leqslant}\,q 0\), and normalizing constant Cq. These distributions, sometimes referred to as q-Gaussians, are known to make, under appropriate constraints, extremal the functional Sq (in its continuous version). Their q = 1 and q = 2 particular cases recover respectively Gaussian and Cauchy distributions.