Numerical indications of a q-generalised central limit theorem

TL;DR

This paper provides numerical evidence supporting the $q$-generalised central limit theorem in nonextensive statistical mechanics, showing that correlated binary variables tend to $q_e$-Gaussians in the large N limit.

Contribution

It demonstrates numerically that correlated binary variables under specific scale-invariant rules converge to $q_e$-Gaussians, extending the classical CLT to nonextensive contexts.

Findings

Emerging distributions are $q_e$-Gaussians with $q_e=2-rac{1}{q}$.

Coefficients $eta(N)$ approach finite limits as N increases.

Recovers classical CLT when $q=q_e=1$.

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.