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In order to physically enlighten the relationship between {\it$q$--independence} and {\it scale-invariance}, we introduce three types ofasymptotically scale-invariant probabilistic models with binary randomvariables, namely (i) a family, characterized by an index $\nu=1,2,3,...$,unifying the Leibnitz triangle ($\nu=1$) and the case of independent variables($\nu\to\infty$); (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$\chi=1/2$). Models (i) and (iii) are in fact strictly scale-invariant. Formodels (i), we analytically show that the $N \to\infty$ probabilitydistribution is a $q$--Gaussian with $q=(\nu -2)/(\nu-1)$. Models (ii) approach$q$--Gaussians by construction, and we numerically show that they do so withasymptotic scale-invariance. Models (iii), like two other strictlyscale-invariant models recently discussed by Hilhorst and Schehr (2007),approach instead limiting distributions which are {\it not} $q$--Gaussians. Thescenario which emerges is that asymptotic (or even strict) scale-invariance isnot sufficient but it might be necessary for having strict (or asymptotic)$q$--independence, which, in turn, mandates $q$--Gaussian attractors.

journal of statistical mechanics

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