  We introduce a family of probabilistic {\it scale-invariant} Leibniz-like
pyramids and $(d+1)$-dimensional hyperpyramids ($d=1,2,3,...$), characterized
by a parameter $\nu>0$, whose value determines the degree of correlation
between $N$ $(d+1)$-valued random variables. There are $(d+1)^N$ different
events, and the limit $\nu\to\infty$ corresponds to independent random
variables, in which case each event has a probability $1/(d+1)^N$ to occur. The
sums of these $N$ $\,(d+1)$-valued random variables correspond to a
$d-$dimensional probabilistic model, and generalizes a recently proposed
one-dimensional ($d=1$) model having $q-$Gaussians (with $q=(\nu-2)/(\nu-1)$
for $\nu \in [1,\infty)$) as $N\to\infty$ limit probability distributions for
the sum of the $N$ binary variables [A. Rodr\'{\i}guez {\em et al}, J. Stat.
Mech. (2008) P09006; R. Hanel {\em et al}, Eur. Phys. J. B {\bf 72}, 263
(2009)]. In the $\nu\to\infty$ limit the $d-$dimensional multinomial
distribution is recovered for the sums, which approach a $d-$dimensional
Gaussian distribution for $N\to\infty$. For any $\nu$, the conditional
distributions of the $d-$dimensional model are shown to yield the corresponding
joint distribution of the $(d-1)$-dimensional model with the same $\nu$. For
the $d=2$ case, we study the joint probability distribution, and identify two
classes of marginal distributions, one of them being asymmetric and
scale-invariant, while the other one is symmetric and only asymptotically
scale-invariant. The present probabilistic model is proposed as a testing
ground for a deeper understanding of the necessary and sufficient conditions
for having $q$-Gaussian attractors in the $N\to\infty$ limit, the ultimate goal
being a neat mathematical view of the causes clarifying the ubiquitous
emergence of $q$-statistics verified in many natural, artificial and social
systems.
