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

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 \(1{\leqslant}\,q < 3\). The attractor, in the usual sense of a central limit theorem, is given by a distribution of the form \(p(x) = C_{q}[1 - (1 - q)\beta x^{2}]^{1/(1-q)} {\rm with} \beta > 0\), and normalizing constant C _q . These distributions, sometimes referred to as q-Gaussians, are known to make, under appropriate constraints, extremal the functional S _q (in its continuous version). Their q = 1 and q = 2 particular cases recover respectively Gaussian and Cauchy distributions.