Strictly and asymptotically scale-invariant probabilistic models of $N$ correlated binary random variables having {\em q}--Gaussians as $N\to \infty$ 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 $ν=1,2,3,...$, unifying the Leibnitz triangle ($ν=1$) and the case of independent variables ($ν\to\infty$); (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 $N \to\infty$ 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 (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.