Strictly and asymptotically scale invariant probabilistic models ofN correlated binaryrandom variables having q-Gaussians as N → ∞ limiting distributions

Abstract

The celebrated Leibnitz triangle has a remarkable property, namely that eachof its elements equals the sum of its south-west and south-east neighbors.In probabilistic terms, this corresponds to a specific form of correlation ofN equally probable binary variables which satisfy scale invariance. Indeed, the marginal probabilities ofthe N-system precisely coincide with the joint probabilities of the(N−1)-system. On the other hand, the non-additive entropy , which grounds non-extensive statistical mechanics, is, under appropriate constraints, extremized bythe (q-Gaussian) distribution (q<3; ). These distributions also result, as attractors, from a generalized central limit theorem for randomvariables which have a finite generalized variance, and are correlated in a specific way calledq-independence. In order to provide physical enlightenment as regards this concept, weintroduce here three types of asymptotically scale invariant probabilistic modelswith binary random variables, namely (i) a family, characterized by an indexν = 1,2,3,..., unifying theLeibnitz triangle (ν = 1) and the case of independent variables (); (ii) two slightly different discretizations ofq-Gaussians; (iii) a special family, characterized by the parameterχ, which generalizes the usual case of independent variables (recovered forχ = 1/2). Models (i) and (iii) are in fact strictly scale invariant. For models (i), we analytically showthat the N → ∞ probability distribution is a q-Gaussian with q = (ν−2)/(ν−1).Models (ii) approach q-Gaussians by construction, and we numerically show that they do so with asymptotic scaleinvariance. Models (iii), like two other strictly scale invariant models recentlydiscussed by Hilhorst and Schehr, approach instead limiting distributions which arenot q-Gaussians. The scenario which emerges is that asymptotic (or even strict) scaleinvariance is not sufficient but it might be necessary for having strict (or asymptotic)q-independence, which,in turn, mandates q-Gaussian attractors.