Beyond the Central Limit: Universality of the Gamma Distribution from Padé-Enhanced Large Deviations

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.

Figures (3)

• Figure 1: Empirical distributions of the sum $S_n$ of $n=30$ exponential variables with rates drawn from various hyper-distributions $q(\lambda)$. (a) Uniform[1,2]; (b) Log-normal(1,2); (c) power-law $q(\lambda)\sim\lambda^{-1}$; (d) Uniform[0,1]; (e) One outlier rate (see text); (f) KL divergence difference between normal and gamma. The dashed straight line scales as $\sim 1/n$. The gamma approximation (orange line) consistently outperforms the normal approximation (blue line). In the title, the KL divergence difference is always positive (the gamma is better). The cases in the bottom row show some discrepancies in the tails, due to the presence of some rate(s) $\lambda_i\simeq 0$ (see text).
• Figure 2: Empirical distribution of $S_{15}$ and the corresponding normal (blue diamonds), gamma (orange triangles), and shifted-gamma (purple circles) approximations for the sum of variables drawn from a truncated normal distribution with different values of $\mu$. a) $\mu/\sigma=-1$; b) $\mu/\sigma=1$; c) $\mu/\sigma=2$; d) Numerical KLD (symbols) and approximation Eq. \ref{['eq:kappa3']}, for the normal and the gamma, and Eq. \ref{['eq:kappa4']} for the shifted gamma. Note how at $\mu/\sigma\sim 1$ (orange dashed line for the gamma, and purple dashed for the shifted gamma), both KLD cross, meaning that above that, the normal approximates better the empirical distribution, and below, the gamma is. Also note how Eqs. \ref{['eq:kappa3']}-\ref{['eq:kappa4']} capture the transition qualitatively. In the case of the shifted gamma, the approximation extends up to $\mu/\sigma\simeq 2$ as the third cumulant can also be matched (see Supplemental Material, Table S2).
• Figure 3: Gamma (orange) and normal (blue) approximations of the empirical histogram for a single (triangles) generalized gamma ($\theta=1.75$) and the sum (circles) of generalized gamma random variables from $n=6$ ecological niches camacho2025microbial with parameters $\lambda=1$, $\theta_1,\ldots,\theta_6=1.49, 1.54, 1.11, 1.60, 1.75, 1.42$, respectively, and a) $k=0.50$ and b) $k=3$.