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

Abstract

We study a symmetric generalization pk(N)(η,α) of the binomial distribution recently introduced by Bergeron et al., where η ∈ [0, 1] denotes the win probability and α is a positive parameter. This generalization is based on q-exponential generating functions (eqgenz≡[1+(1−qgen)z]1/(1−qgen); e1z=ez) where qgen = 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 qdisc-Gaussian distribution, for win-loss equiprobability (i.e., η = 1/2) and all values of α. Asymptotic N → ∞ distribution is in fact a qatt-Gaussian eqatt−βz2, where qatt = 1 − 2/(α − 2) and β = (2α − 4). The behavior of the scaled quantity k/Nγ is discussed as well. For γ < 1, a large-deviation-like property showing a qldl-exponential decay is found, where qldl = 1 + 1/(ηα). For η = 1/2, qldl and qatt are related through 1/(qldl − 1) + 1/(qatt − 1) = 1, ∀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 → ∞ limit distribution, yielding a power law, although not exactly a qLD-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.