, Volume 76, Issue 1, pp 307328
First online:
On a qCentral Limit Theorem Consistent with Nonextensive Statistical Mechanics
 Sabir UmarovAffiliated withDepartment of Mathematics, Tufts University Email author
 , Constantino TsallisAffiliated withCentro Brasileiro de Pesquisas FisicasSanta Fe Institute
 , Stanly SteinbergAffiliated withDepartment of Mathematics and Statistics, University of New Mexico
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Get AccessAbstract.
The standard central limit theorem plays a fundamental role in BoltzmannGibbs statistical mechanics. This important physical theory has been generalized [1] in 1988 by using the entropy \(S_{q} = \frac{1\sum_{i} p^{q}_{i}}{q1}\) \(({\rm with}\,q\,\in {{{\mathcal{R}}}})\) 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 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 qversions 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 \(p(x) = C_{q}[1  (1  q)\beta x^{2}]^{1/(1q)} {\rm with} \beta > 0\), and normalizing constant C _{ q }. These distributions, sometimes referred to as qGaussians, 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.
Mathematics Subject Classification (2000).
Primary 60F05 Secondary 60E07, 60E10, 82CxxKeywords.
qcentral limit theorem correlated random variables nonextensive statistical mechanics Title
 On a qCentral Limit Theorem Consistent with Nonextensive Statistical Mechanics
 Journal

Milan Journal of Mathematics
Volume 76, Issue 1 , pp 307328
 Cover Date
 200812
 DOI
 10.1007/s000320080087y
 Print ISSN
 14249286
 Online ISSN
 14249294
 Publisher
 BirkhäuserVerlag
 Additional Links
 Topics
 Keywords

 Primary 60F05
 Secondary 60E07, 60E10, 82Cxx
 qcentral limit theorem
 correlated random variables
 nonextensive statistical mechanics
 Authors

 Sabir Umarov ^{(1)}
 Constantino Tsallis ^{(2)} ^{(3)}
 Stanly Steinberg ^{(4)}
 Author Affiliations

 1. Department of Mathematics, Tufts University, Medford, MA, 02155, USA
 2. Centro Brasileiro de Pesquisas Fisicas, Xavier Sigaud 150, 22290180, Rio de JaneiroRJ, Brazil
 3. Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM, 87501, USA
 4. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM, 87131, USA