  The correlated probabilistic model introduced and analytically discussed in
Hanel et al (2009) is based on a self-dual transformation of the index $q$
which characterizes a current generalization of Boltzmann-Gibbs statistical
mechanics, namely nonextensive statistical mechanics, and yields, in the
$N\to\infty$ limit, a $Q$-Gaussian distribution for any chosen value of $Q \in
[1,3)$. We show here that, by properly generalizing that self-dual
transformation, it is possible to obtain an entire family of such probabilistic
models, all of them yielding $Q_c$-Gaussians ($Q_c \ge 1$) in the $N\to\infty$
limit. This family turns out to be isomorphic to the Hanel et al model through
a specific monotonic transformation $Q_c(Q)$. Then, by following along the
lines of Tirnakli et al (2022), we numerically show that this family of
correlated probabilistic models provides further evidence towards a
$q$-generalized Large Deviation Theory (LDT), consistently with the Legendre
structure of thermodynamics. The present analysis deepens our understanding of
complex systems (with global correlations among their elements), supporting the
conjecture that generic models whose attractors under summation of $N$
strongly-correlated random variables are $Q$-Gaussians, might always be
concomitantly associated with $q$-exponentials in the LDT sense.
