Abstract
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 
, which grounds non-extensive statistical mechanics, is, under appropriate constraints, extremized by
the (q-Gaussian) distribution 
(q<3; 
). 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 (
); (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.