[Paper Review] 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 Analysis0 citations

TL;DR

gamma distributions emerge from a Padé-enhanced large deviation framework for sums of positive random variables, providing a universal mechanism beyond the central limit theorem and preserving positivity.

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.

Motivation & Objective

• 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.

Proposed method

• 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.

Experimental results

Research questions

• RQ1 When does the gamma distribution arise as a universal limit for sums of positive or heterogeneous random variables?
• RQ2 How does the Padé-enhanced large deviation approach compare to the normal approximation across diverse distributions (exponential, truncated normal, generalized gamma) in terms of accuracy?
• RQ3 Can the Padé framework be extended to handle convolutions of gamma distributions and non-Markovian aggregation processes?
• RQ4 What practical guidance does the approach provide for modeling empirical positive data where positivity and constraints are important?

Key findings

• The gamma density p_n^(G)(x) = c_n (x/μ_n)^{α_n-1} exp(-α_n x/μ_n) with α_n = μ_n^2/σ_n^2 naturally emerges from the [0/1] Padé approximant within LDT.
• For sums of independent non-identical exponential variables, the gamma approximation is exact when variables are i.i.d. exponential and often superior to the normal approximation in KL divergence for broad cases.
• Across non-identical exponential sums, truncated normal sums, and non-identical generalized gamma sums, the gamma approximation consistently outperforms the normal approximation in empirical tests (lower KL divergence).
• Positivity constraints are preserved by the Padé approach, and the method yields shifted gamma variants (e.g., [1/1] Padé) that can further improve tail behavior.
• The framework accommodates extensions to convolutions of gamma distributions and non-Markovian dynamics, offering a general methodological tool for constrained aggregation.