A generalization of the 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 \cite{Tsallis1988} in 1988 by using the entropy $S_q = \frac{1-\sum_i p_i^q}{q-1}$ (with $q \in \mathcal{R}$) instead of its particular BG case $S_1=S_{BG}= -\sum_i p_i \ln p_i$. The theory which emerges is usually referred to as {\it 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\cite{Tsallis2005} 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 $1 \leq q<3$. 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 $C_q$. 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.