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

We study a symmetric generalization $p k ( N ) ( η , α )$ 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 $( e q gen z ≡ [ 1 + ( 1 − q gen ) z ] 1 / ( 1 − q gen ) ; e 1 z = e z )$ 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 → ∞ distribution is in fact a q^att-Gaussian $e q att − β z 2$⁠, 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, ∀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 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.