Central limit theorem for a class of globally correlated random variables

TL;DR

This paper extends the central limit theorem to globally correlated random variables, revealing new classes of limit distributions influenced by correlation mechanisms, including q-Gaussian attractors.

Contribution

It derives hierarchical equations for correlated variables, defining classes of memory mechanisms and characterizing their limit distributions, including urn models and q-Gaussians.

Findings

Different correlation mechanisms lead to distinct limit distributions.

Urn models' domains of attraction are characterized by variable densities.

Symmetric and asymmetric q-Gaussian attractors are identified as special cases.

Abstract

The standard central limit theorem with a Gaussian attractor for the sum of independent random variables may lose its validity in presence of strong correlations between the added random contributions. Here, we study this problem for similar interchangeable globally correlated random variables. Under these conditions, a hierarchical set of equations is derived for the conditional transition probabilities. This result allows us to define different classes of memory mechanisms that depend on a symmetric way on all involved variables. Depending on the correlation mechanisms and single statistics, the corresponding sums are characterized by distinct statistical probability densities. For a class of urn models it is also possible to characterize their domain of attraction which, as in the standard case, is parametrized by the probability density of each random variable. Symmetric and asymmetric $q$-Gaussian attractors $(q<1)$ are a particular case of these models.