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

We study a symmetric generalization $\mathfrak{p}^{(N)}_k(η, α)$ of the binomial distribution recently introduced by Bergeron et al, where $η\in [0,1]$ denotes the win probability, and $α$ is a positive parameter. This generalization is based on $q$-exponential generating functions ($e_{q^{gen}}^z \equiv [1+(1-q^{gen})z]^{1/(1-q^{gen})};\,e_{1}^z=e^z)$ where $q^{gen}=1+1/α$. 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., $η=1/2$) and all values of $α$. Asymptotic $N\to \infty$ distribution is in fact a $q^{att}$-Gaussian $e_{q^{att}}^{-βz^2}$, where $q^{att}=1-2/(α-2)$ and $β=(2α-4)$. The behavior of the scaled quantity $k/N^γ$ is discussed as well. For $γ<1$, a large-deviation-like property showing a $q^{ldl}$-exponential decay is found, where $q^{ldl}=1+1/(ηα)$. For $η=1/2$, $q^{ldl}$ and $q^{att}$ are related through $1/(q^{ldl}-1)+1/(q^{att}-1)=1$, $\forall α$. For $γ=1$, the law of large numbers is violated, and we consistently study the large-deviations with respect to the probability of the $N\to\infty$ limit distribution, yielding a power law, although not exactly a $q^{LD}$-exponential decay. All $q$-statistical parameters which emerge are univocally defined by $(η, α)$. Finally we discuss the analytical connection with the Pólya urn problem.