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 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 qe-Gaussians, i.e., p(x) ∝ [1-(1-q_e), β(N)x^2]^{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.

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