Milan Journal of Mathematics

, Volume 76, Issue 1, pp 307–328

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

• Sabir Umarov
• Constantino Tsallis
• Stanly Steinberg
Article

DOI: 10.1007/s00032-008-0087-y

Umarov, S., Tsallis, C. & Steinberg, S. Milan j. math. (2008) 76: 307. doi:10.1007/s00032-008-0087-y

## Abstract.

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 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

### Keywords.

q-central limit theorem correlated random variables nonextensive statistical mechanics

## Authors and Affiliations

• Sabir Umarov
• 1
• Constantino Tsallis
• 2
• 3
• Stanly Steinberg
• 4
1. 1.Department of MathematicsTufts UniversityMedfordUSA
2. 2.Centro Brasileiro de Pesquisas FisicasRio de Janeiro-RJBrazil
3. 3.Santa Fe InstituteSanta FeUSA
4. 4.Department of Mathematics and StatisticsUniversity of New MexicoAlbuquerqueUSA