Strictly and asymptotically scale-invariant probabilistic models of $N$ correlated binary random variables having {\em q}--Gaussians as $N\to \mathrm{\infty }$ limiting distributions

In order to physically enlighten the relationship between {\it $q$–independence} and {\it scale-invariance}, we introduce three types of asymptotically scale-invariant probabilistic models with binary random variables, namely (i) a family, characterized by an index $\nu =1,2,3,\mathrm{.}\mathrm{.}\mathrm{.}$, unifying the Leibnitz triangle ($\nu =1$) and the case of independent variables ($\nu \to \mathrm{\infty }$); (ii) two slightly different discretizations of $q$–Gaussians; (iii) a special family, characterized by the parameter $\chi$, which generalizes the usual case of independent variables (recovered for $\chi =1\mathrm{/}2$). Models (i) and (iii) are in fact strictly scale-invariant. For models (i), we analytically show that the $N\to \mathrm{\infty }$ probability distribution is a $q$–Gaussian with $q=(\nu -2)\mathrm{/}(\nu -1)$. 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 (2007), approach instead limiting distributions which are {\it 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.