Numerical indications of a q-generalised central limit theorem

Abstract

We provide numerical indications of the $q$-generalised central limit theorem that has been conjectured (Tsallis 2004) in nonextensive statistical mechanics. We focus on $N$ binary random variables correlated in a {\it 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 $q_e$-Gaussians, i.e., $p(x)\propto [1-(1-q_e)\beta (N)x^2{]}^{1/(1-q_e)}$, with $q_e=2-\frac{1}{q}$, and with coefficients $\beta (N)$ approaching finite values $\beta (\infty )$. The particular case $q=q_e=1$ recovers the celebrated de Moivre-Laplace theorem.

Minor improvements and corrections have been introduced in the new version. 7 pages including 4 figures