A generalization of the central limit theorem consistent with nonextensive statistical mechanics

Abstract

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 $1\le q<3$. 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.

19 pages (the new version contains further simplifications and precisions with regard to the previous one)

References in corpus (17)

• Anomalous diffusion in the presence of external forces: exact time-dependent solutions and entropy
• A possible deformed algebra and calculus inspired in nonextensive thermostatistics
• Generalized algebra within a nonextensive statistics
• Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive
• What should a statistical mechanics satisfy to reflect nature?
• Anomalous diffusion associated with nonlinear fractional derivative Fokker-Planck-like equation: Exact time-dependent solutions
• Defect turbulence and generalized statistical mechanics
• Triangle for the entropic index q of non-extensive statistical mechanics observed by Voyager 1 in the distant heliosphere
• Dynamical scenario for nonextensive statistical mechanics
• Numerical indications of a q-generalised central limit theorem
• Nonextensive Thermodynamics and Glassy behaviour in Hamiltonian systems
• Anomalous diffusion in silo drainage
• Random walk models associated with distributed fractional order differential equations
• Non-extensive random walks
• Influence of global correlations on central limit theorems and entropic extensivity
• Some thoughts on theoretical physics
• Multivariate Generalizations of the q--Central Limit Theorem

Cited by in corpus (22)

• Generalization of symmetric $\alpha$-stable Lévy distributions for $q>1$
• Quasiclassical Coarse Graining and Thermodynamic Entropy
• The q-exponential family in statistical physics
• A closer look at the indications of q-generalized Central Limit Theorem behavior in quasi-stationary states of the HMF model
• Strictly and asymptotically scale-invariant probabilistic models of $N$ correlated binary random variables having {\em q}--Gaussians as $N\to \infty$ limiting distributions
• Note on a q-modified central limit theorem
• Nonextensive statistical mechanics and central limit theorems I - Convolution of independent random variables and q-product
• Tsallis non-extensive statistics, intermittent turbulence, SOC and chaos in the solar plasma. Part one: Sunspot dynamics
• Fractional Fokker-Planck equation from non-singular kernel operators
• On a representation of the inverse Fq transform
• On the average uncertainty for systems with nonlinear coupling
• On the role of ergodicity and mixing in the central limit theorem for Casati-Prosen triangle map variables
• Open Problems within Nonextensive Statistical Mechanics
• The limit distribution in the $q$-CLT for $q\ge 1$ is unique and can not have a compact support
• Multivariate Generalizations of the q--Central Limit Theorem
• A definition of the coupled-product for multivariate coupled-exponentials
• A scale-invariant probabilistic model based on Leibniz-like pyramids
• Emergence of $q$-statistical functions in a generalized binomial distribution with strong correlations
• The ${\tau }_q$-Fourier transform: covariance and uniqueness
• On a Connection between Entropy, Extensive Measurement and Memoryless Characterization
• Limit distribution in the $q$-CLT for $q\ge 1$ can not have a compact support
• On the Gaussian q-Distribution