Rodríguez Mesas, Antonio and Schwämmle, Veit and Tsallis, Constantino
Strictly and asymptotically scale invariant probabilistic models of N correlated binary random variables having q-Gaussians as N → ∞ limiting distributions.
"Journal of Statistical Mechanics: Theory and Experiment", v. 2008
The celebrated Leibnitz triangle has a remarkable property, namely that each of its elements equals the sum of its south-west and south-east neighbors. In probabilistic terms, this corresponds to a specific form of correlation of N equally probable binary variables which satisfy scale invariance. Indeed, the marginal probabilities of the N-system precisely coincide with the joint probabilities of the (N − 1)-system. On the other hand, the non-additive entropy Sq ≡ (1 − ∫_∞^∞▒〖p(x)]q)/(q - 1)〗 (q ∈ R; S1 = −∫_∞^∞▒〖p(x) ln p(x)〗), which grounds non-extensive statistical mechanics, is, under appropriate constraints, extremized by the (q-Gaussian) distribution pq(x) ∝ [1 − (1 − q)β x2]1/(1−q) (q < 3; p1(x) ∝ e−βx2 ). These distributions also result, as attractors, from a generalized central limit theorem for random variables which have a finite generalized variance, and are correlated in a specific way called q-independence. In order to provide physical enlightenment as regards this concept, we introduce here three types of asymptotically scale invariant probabilistic models with binary random variables, namely (i) a family, characterized by an index ν = 1, 2, 3, . . ., unifying the Leibnitz triangle (ν = 1) and the case of independent variables (ν →∞); (ii) two slightly different discretizations of q-Gaussians; (iii) a special family, characterized by the parameter χ, which generalizes the usual case of independent variables (recovered for χ = 1/2). Models (i) and (iii) are in fact strictly scale invariant. For models (i), we analytically show that 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 scale invariance. Models (iii), like two other strictly scale invariant models recently discussed by Hilhorst and Schehr, approach instead limiting distributions which are not q-Gaussians. The scenario which emerges is that asymptotic (or even strict) scale invariance is not sufficient but it might be necessary for having strict (or asymptotic) q-independence, which, in turn, mandates q-Gaussian attractors.