A Generalization of the Central Limit Theorem Consistent with Nonextensive Statistical Mechanics

As well known, the standard central limit theorem plays a fundamental role in BoltzmannGibbs (BG) statistical mechanics. This important physical theory has been generalized by one of us (CT) in 1988 by using the entropy Sq = 1− ∑ i p i q−1 (with q ∈ R) instead of its particular case S1 = SBG = − ∑ i pi ln pi. 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. Conjectures and numerical indications available in the literature were since a few years suggesting the possibility of q-generalizations of the standard central limit theorem by allowing the random variables that are being summed to be correlated in some special manner, the case q = 1 corresponding to standard probabilistic independence. This is precisely what we prove in the present paper for some range of q which extends from below to above q = 1. The attractor, in the usual sense of a central limit theorem, is given by a distribution of the form p(x) ∝ [1−(1−q)βx2]1/(1−q) with β > 0. These distributions, sometimes referred to as q-Gaussians, are known to make, under appropriate constraints, extremal the functional Sq. Their q = 1 and q = 2 particular cases recover respectively Gaussian and Cauchy distributions.