A Constructive Approach to $q$-Gaussian Distributions: $\alpha$-Divergence as Rate Function and Generalized de Moivre-Laplace Theorem

TL;DR

This paper develops a constructive probabilistic framework for power-law distributions, deriving generalized binomial and $q$-Gaussian distributions, establishing large deviation principles, and connecting to information geometry.

Contribution

It introduces a novel constructive approach to power-law distributions from nonlinear differential equations, deriving generalized binomial and $q$-Gaussian distributions with new theoretical insights.

Findings

Proves the LDP for the generalized binomial distribution with $0<q<1$

Identifies $ ext{alpha}$-divergence as the rate function

Shows convergence to $q$-Gaussian distribution with $n^{q/2}$ scaling

Abstract

The Large Deviation Principle (LDP) and the Central Limit Theorem (CLT) are central pillars of probability theory. While their formulations are established under the i.i.d. assumption, the probabilistic foundation for power-law distributions has primarily evolved through descriptive models or variational principles, rather than a constructive derivation comparable to the classical binomial process. This paper establishes a constructive probabilistic framework for power-law distributions, proceeding from the nonlinear differential equation $dy/dx = y^q$ without assuming a specific distribution a priori. We build the algebraic and combinatorial foundations, which lead to a generalized binomial distribution based on finite counting. We prove the LDP for this generalized binomial distribution in the regime $0 < q < 1$, demonstrating that the $\alpha$-divergence is identified as the rate function, and clarify the breakdown of this macroscopic scaling for heavier tails ($q > 1$). This result connects our constructive framework to the structures of information geometry. Furthermore, we prove a generalized de Moivre-Laplace theorem, showing that the generalized binomial distribution converges to a heavy-tailed limit distribution (the $q$-Gaussian distribution). We derive that the scaling law follows the order of $n^{q/2}$ as a consequence of the underlying nonlinearity. These analytical results are numerically verified for distinct values of $q \in (0, 2)$. This framework provides a constructive basis that unifies the shift-invariant exponential family and the rescaling-invariant power-law family.