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

摘要

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.