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 2004) 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 − qe) β(N)x ]e, with qe = 2 − 1 q , and with coefficients β(N) approaching finite values β(∞). The particular case q = qe = 1 recovers the celebrated de Moivre-Laplace theorem. Introduction. – The central limit theorem (CLT) is a cornerstone of probability theory and is of fundamental importance in statistical mechanics. This important theorem implies, roughly speaking, that any sum of N independent random variables will tend, as N → ∞, to be distributed according to a certain law (which behaves as an attractor in the space of distributions). When the distribution of the individual random variables has any finite variance, the attractor for the sum will be a normal (Gaussian) distribution [1], and this is the result usually known as CLT (from now on denoted G-CLT). Several extensions of the CLT exist, such as the one due to Gnedenko, Kolmogorov, and Lévy [1] (from now on denoted L-CLT), widely known in physics because of its relation with anomalous diffusion [2]. This extension states that the sum of independent infinite-variance variables will be attracted to Lévy distributions. The G-CLT explains the frequent occurrence of normal distributions in nature. Its first manifestation in mathematics was due to Abraham de Moivre in 1733, followed independently by Pierre-Simon de Laplace in 1774. The distribution was rediscovered by Robert Adrain in 1808, and then finally by Carl Friedrich Gauss, who based on it his famous theory of errors [3]. A central result is the fact that the binomial distribution approaches, for N → ∞ and after being appropriately centralised and rescaled, a Gaussian. This can be considered as the first historical manifestation of the G-CLT. It is frequently referred to as the de Moivre-Laplace