Influence of global correlations on central limit theorems and entropic extensivity

TL;DR

This paper investigates how global correlations in probabilistic models affect the central limit theorem and entropy extensivity, revealing conditions under which non-Gaussian attractors and nonextensive entropies emerge.

Contribution

It provides new insights into the impact of global correlations on limit theorems and entropy, illustrating scenarios where non-Gaussian attractors and q-entropies become relevant.

Findings

Global correlations can alter the Gaussian attractor in the CLT.

Entropy remains extensive under certain strong correlations.

q-entropy can be extensive for q ≠ 1 in correlated systems.

Abstract

We consider probabilistic models of N identical distinguishable, binary random variables. If these variables are strictly or asymptotically independent, then, for N>>1, (i) the attractor in distribution space is, according to the standard central limit theorem, a Gaussian, and (ii) the Boltzmann-Gibbs-Shannon entropy is extensive, meaning that S_BGS(N) ~ N . If these variables have any nonvanishing global (i.e., not asymptotically independent) correlations, then the attractor deviates from the Gaussian. The entropy appears to be more robust, in the sense that, in some cases, S_BGS remains extensive even in the presence of strong global correlations. In other cases, however, even weak global correlations make the entropy deviate from the normal behavior. More precisely, in such cases the entropic form Sq can become extensive for some value of q different from unity . This scenario is illustrated with several new as well as previously described models. The discussion illuminates recent progress into q-describable nonextensive probabilistic systems, and the conjectured q-Central Limit Theorem (q-CLT) which posses a q-Gaussian attractor.