The Large Deviation Principle (LDP) and the Central Limit Theorem (CLT) are concepts of information theory and probability. 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）和中心极限定理（CLT）是信息论和概率论的核心概念。尽管它们的标准描述建立在独立同分布（i.i.d.）假设之上，但幂律分布的概率基础主要经由描述性模型或变分原理发展而来，缺少能与经典二项过程相媲美的构造性推导。本文为幂律分布建立了一个构造性概率框架，从非线性微分方程$dy/dx = y^q$出发，无需预先假定特定分布。我们构建了代数和组合基础，进而推导出基于有限计数的广义二项分布。在$0 < q < 1$范围内，我们证明了该广义二项分布的LDP，并表明$α$-散度被识别为率函数，同时阐明了对于更重尾分布（$q > 1$）此宏观标度失效的原因。这一结果将我们的构造框架与信息几何结构联系起来。此外，我们证明了一个广义的棣莫弗-拉普拉斯定理，展示了广义二项分布收敛于重尾极限分布（即$q$-高斯分布）。我们推导出标度律遵循$n^{q/2}$阶次，这是底层非线性的直接结果。针对$q \in (0, 2)$的不同取值，这些分析结果均得到了数值验证。该框架提供了一种统一平移不变指数族和重标度不变幂律族的构造性基础。