Emergence of $q$-statistical functions in a generalized binomial distribution with strong correlations

We study a symmetric generalization ${\mathfrak{p}}_k^{(N)}(\eta ,\alpha )$ of the binomial distribution recently introduced by Bergeron et al, where $\eta \in [0,1]$denotesthewinprobability,and$\alpha$ is a positive parameter. This generalization is based on $q$-exponential generating functions ($e_{q^{gen}}^z\equiv [1+(1-q^{gen})z{]}^{1\mathrm{/}(1-q^{gen})};e_1^z=e^z)$ where $q^{gen}=1+1\mathrm{/}\alpha$. The numerical calculation of the probability distribution function of the number of wins $k$, related to the number of realizations $N$, strongly approaches a discrete $q^{disc}$-Gaussian distribution, for win-loss equiprobability (i.e., $\eta =1\mathrm{/}2$) and all values of $\alpha$. Asymptotic $N\to \infty$distributionisinfacta$q^{att}$-Gaussian$e_{q^{att}}^{-\beta z^2}$,where$q^{att}=1-2/(\alpha-2)$and$\beta=(2\alpha-4)$. The behavior of the scaled quantity $k\mathrm{/}{N}^{\gamma }$ is discussed as well. For $\gamma <1$, a large-deviation-like property showing a $q^{ldl}$-exponential decay is found, where $q^{ldl}=1+1\mathrm{/}(\eta \alpha )$. For $\eta =1\mathrm{/}2$, $q^{ldl}$ and $q^{att}$ are related through $1\mathrm{/}(q^{ldl}-1)+1\mathrm{/}(q^{att}-1)=1$, $\mathrm{\forall }\alpha$. For $\gamma =1$, the law of large numbers is violated, and we consistently study the large-deviations with respect to the probability of the $N\to \mathrm{\infty }$ limit distribution, yielding a power law, although not exactly a $q^{LD}$-exponential decay. All $q$-statistical parameters which emerge are univocally defined by $(\eta ,\alpha )$. Finally we discuss the analytical connection with the P'{o}lya urn problem.