[论文解读] Beyond the Central Limit: Universality of the Gamma Distribution from Padé-Enhanced Large Deviations

Mario Castro, José A. Cuesta|arXiv (Cornell University)|Mar 24, 2026

Earthquake Detection and Analysis被引用 0

一句话总结

伽马分布从对正变量和多样化样本和大偏差框架中的 Padé 改进的和 sum 的框架中涌现，提供一种超越中心极限定理、保持正性的一般机制。

ABSTRACT

The central limit theorem provides the theoretical foundation for the universality of the normal distribution: under broad conditions, the asymptotic distribution of a sum of independent random variables approaches a Gaussian. Yet, physical systems described by positive random variable -- from earthquakes to microbial growth to epidemic spreading -- consistently exhibit gamma rather than Gaussian statistics -- what leads to field-specific mechanistic explanations that are non robust to small changes in the model details. We show that gamma distributions emerge naturally from large deviation theory when Padé approximants replace polynomial expansions of the derivative of the scaled cumulant generating function, respecting positivity constraints that the central limit theorem violates. Gamma universality thus emerges as the constrained analog of Gaussian universality, providing a mechanism-free explanation for its pervasive appearance across different disciplines.

研究动机与目标

• Motivate the limitations of the central limit theorem for positive, heterogeneous, or small-sample sums.
• Introduce a Padé approximant to the scaled cumulant generating function within large deviation theory.
• Derive a gamma-type universal density and identify conditions under which it is exact or superior to Gaussian approximations.
• Demonstrate across scenarios (exponentials, truncated normals, generalized gamma) and discuss extensions to convolutions and non-Markovian dynamics.

提出的方法

• Represent the distribution via Laplace transform of the constraint S_n(X)=x.
• Define the scaled CGF lambda_n(ξ) and apply a [0/1] Padé approximation to its derivative: n lambda_n'(ξ) ≈ μ_n/(1 - σ_n^2 ξ/μ_n).
• Integrate to obtain n lambda_n(ξ) and perform saddle-point evaluation to derive a gamma-like density.
• Show that the resulting p_n^(G)(x) has the form c_n (x/μ_n)^{α_n-1} exp(-α_n x/μ_n) with α_n = μ_n^2/σ_n^2.
• Argue positivity (ξ < μ_n/σ_n^2) and contrast with the CLT; discuss accuracy via KL divergence and Edgeworth-like reasoning.
• Present extensions to shifted gamma, higher-order Padé (leading to convolutions of shifted gammas) and non-Markovian dynamics.

实验结果

研究问题

• RQ1 何时伽马分布作为正变量或异质随机变量和/或小样本和的普遍极限出现？
• RQ2 在不同分布（指数、截断正态、广义伽马）下，Padé 增强的大偏差方法相对于正态逼近在精度方面有何比较？
• RQ3 Padé 框架是否可以扩展以处理伽马分布的卷积和非马尔科夫聚合过程？
• RQ4 对于对正性和约束重要的经验正数据建模，该方法能提供哪些实际指导？

主要发现

• 伽马密度 p_n^(G)(x) = c_n (x/μ_n)^{α_n-1} exp(-α_n x/μ_n) 且 α_n = μ_n^2/σ_n^2，自然从 LDT 内的 [0/1] Padé 近似中涌现。
• 当变量为独立非同分布的指数变量时，在变量为同分布指数且大多数情况下 KL 散度相对于正态近似更优时，伽马近似是精确的。
• 在非同分布指数和截断正态和非同分布广义伽马和的和中，伽马近似在经验测试中始终优于正态近似（KL 散度较低）。
• Padé 方法保持了正性约束，并且该方法产生平移伽马变体（如 [1/1] Padé），可进一步改善尾部行为。
• 该框架可扩展至伽马分布卷积和非马尔科夫动态，为受约束聚合提供通用方法论工具。