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 $α$-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.

翻译：大偏差原理（LDP）与中心极限定理（CLT）是概率论的核心支柱。尽管其表述建立在独立同分布（i.i.d.）假设下，但幂律分布的概率基础主要通过描述性模型或变分原理发展而来，而非类似于经典二项过程那样通过构造性推导获得。本文在不预先假设特定分布的前提下，从非线性微分方程$dy/dx = y^q$出发，为幂律分布建立了一个构造性概率框架。我们构建了代数与组合学基础，基于有限计数导出了一种广义二项分布。在$0 < q < 1$情形下，我们证明了该广义二项分布满足大偏差原理，并指出$α$-散度被识别为速率函数，同时阐明了当尾部更重（$q > 1$）时该宏观标度关系的失效。这一结果将我们的构造性框架与信息几何结构联系起来。此外，我们证明了广义棣莫弗-拉普拉斯定理，表明该广义二项分布收敛于一个重尾极限分布（$q$-高斯分布）。我们推导出标度律服从$n^{q/2}$阶次，这源于内在的非线性特性。这些解析结果针对$q \in (0, 2)$的不同取值进行了数值验证。该框架提供了一个兼具构造性的基础，统一了平移不变的指数族与重标标度不变的幂律族。