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

Abstract

We study a symmetric generalization ${\mathfrak{p}}_k^{(N)}(\eta ,\alpha )$ of the binomial distribution recently introduced by Bergeron et al, where $\eta \in [0,1]$ denotes the win probability, and $\alpha$ is a positive parameter. This generalization is based on $q$-exponential generating functions ($e_{q^{gen}}^z\equiv [1+(1-q^{gen})z{]}^{1/(1-q^{gen})};e_1^z=e^z)$ where $q^{gen}=1+1/\alpha$. 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., $\eta =1/2$) and all values of $\alpha$. Asymptotic $N\to \infty$ distribution is in fact a $q^{att}$-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 $k/N^{\gamma }$ is discussed as well. For $\gamma <1$, a large-deviation-like property showing a $q^{ldl}$-exponential decay is found, where $q^{ldl}=1+1/(\eta \alpha )$. For $\eta =1/2$, $q^{ldl}$ and $q^{att}$ are related through $1/(q^{ldl}-1)+1/(q^{att}-1)=1$, $\forall \alpha$. For $\gamma =1$, the law of large numbers is violated, and we consistently study the large-deviations with respect to the probability of the $N\to \infty$ 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 Pólya urn problem.

arXiv comment13 pages, 14 figures

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