Europhys. Lett.
Volume 73, Number 6, March 2006
Page(s) 813 - 819
Section General
Published online 15 February 2006
Europhys. Lett., 73 (6), pp. 813-819 (2006)
DOI: 10.1209/epl/i2005-10487-1

Numerical indications of a $\mth{q}$-generalised central limit theorem

L. G. Moyano1, 2, C. Tsallis1, 2 and M. Gell-Mann2

1  Centro Brasileiro de Pesquisas Físicas - Rua Xavier Sigaud 150 22290-180 Rio de Janeiro-RJ, Brazil
2  Santa Fe Institute - 1399 Hyde Park Road, Santa Fe, NM 87501, USA

received 9 September 2005; accepted in final form 30 January 2006
published online 15 February 2006

We provide numerical indications of the q-generalised central limit theorem that has been conjectured (TSALLIS C., Milan J. Math., 73 (2005) 145) in nonextensive statistical mechanics. We focus on N binary random variables correlated in a scale-invariant way. The correlations are introduced by imposing the Leibnitz rule on a probability set based on the so-called q-product with $q \le 1$. We show that, in the large-N limit (and after appropriate centering, rescaling, and symmetrisation), the emerging distributions are qe-Gaussians, i.e., $p(x) \propto [1-(1-q_e)\, \beta(N) x^2]\ensuremath^{1/(1-q_e)}$, with $q_e=2-\frac{1}{q}$, and with coefficients $\beta(N)$ approaching finite values $\beta(\infty)$. The particular case q=qe=1 recovers the celebrated de Moivre-Laplace theorem.

02.50.Cw - Probability theory.
02.70.Rr - General statistical methods.
05.10.-a - Computational methods in statistical physics and nonlinear dynamics.

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