1 4 N ov 2 00 7 A generalization of the central limit theorem consistent with nonextensive statistical mechan - ics

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 Sq = 1− P i p q i q−1 (with q ∈ R) instead of its particular BG case S1 = SBG = − P 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 qgeneralized 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) = Cq[1 − (1 − q)βx ] with β > 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. Mathematics Subject Classification (2000). Primary 60F05; Secondary 60E07, 60E10, 82Cxx.