Influence of global correlations on central limit theorems and entropic extensivity

JOHN A. MARSH; MIGUEL A. FUENTES; LUIS G. MOYANO; C. TSALLIS

PHYSICA A - STATISTICAL AND THEORETICAL PHYSICS

ELSEVIER SCIENCE BV

Lugar: Amsterdam; Año: 2006 vol. 372 p. 183 - 202

0378-4371

We consider probabilistic models of N identical distinguishable, binary random variables. If these variables are strictly or asymptotically independent, then, for N\rightarrow \infty, (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) \sim 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 S_q (with S_1 = S_{BGS}) can become extensive for some value of q\eq 1. 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.