Numerical indications of a q-generalised central limit theorem

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 <= 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) β(N)x²]^{1/(1 - q_e)}, with q_e = 2 - [(1)/(q)], and with coefficients β(N) approaching finite values β(∞). The particular case q = q_e = 1 recovers the celebrated de Moivre-Laplace theorem.