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

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 = 1-Sigma(i)p(i)(q)/q-1 (with q is an element of R) instead of its particular BG case S-1 = S-BG = - Sigma(i)p(i)lnp(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 <= 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)) with beta > 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.