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We study a symmetric generalization $mathfrak{p}^{(N)}_k(eta, alpha)$ 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 $Nto infty$ distribution is in fact a $q^{att}$-Gaussian $e_{q^{att}}^{-beta z^2}$, where $q^{att}=1-2/(alpha-2)$ and $beta=(2alpha-4)$. 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 $q^{ldl}=1+1/(etaalpha)$. 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 $Ntoinfty$ 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.
We generalize some widely used mother wavelets by means of the q-exponential function $e_q^x equiv [1+(1-q)x]^{1/(1-q)}$ ($q in {mathbb R}$, $e_1^x=e^x$) that emerges from nonextensive statistical mechanics. Particularly, we define extend
We calculate the partition function of the $q$-state Potts model on arbitrary-length cyclic ladder graphs of the square and triangular lattices, with a generalized external magnetic field that favors or disfavors a subset of spin values ${1,...,s}$ w
The agent-based Yard-Sale model of wealth inequality is generalized to incorporate exponential economic growth and its distribution. The distribution of economic growth is nonuniform and is determined by the wealth of each agent and a parameter $lamb
We have investigated the proof of the $H$ theorem within a manifestly covariant approach by considering the relativistic statistical theory developed in [Phy. Rev. E {bf 66}, 056125, 2002; {it ibid.} {bf 72}, 036108 2005]. In our analysis, however, w
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