Numerical Indications of a $q$-Generalized Central Limit Theorem

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.