Density Functions and Random Number Generators of $α$-Stable Distributions

2026-06-26

AI Summary

This work addresses the longstanding computational and sampling challenges associated with α-stable distributions, particularly in the multivariate setting, due to the absence of closed-form density expressions. The authors introduce AUB-HTP, a Python toolkit that, for the first time, integrates multiple complementary approaches—including Zolotarev’s integral representation, series expansions, and numerical inversion of the characteristic function—to enable efficient and accurate evaluation of univariate densities. Furthermore, it implements LePage series-based simulation for generating multivariate α-stable random vectors with flexible spectral measures. The proposed toolkit substantially broadens the range of admissible parameters and enhances numerical stability, outperforming existing methods in both accuracy and robustness, thereby providing reproducible and efficient computational support for heavy-tailed modeling.

Abstract

Heavy-tailed distributions are increasingly found to better fit empirical data in engineering, finance, physics, network science, and related fields. Among them, $α$-stable distributions play a central role being limiting laws in the generalized central limit theorem: they are expected to be exceptionally good models whenever sums of multiple independent heavy-tailed sources are at play. Despite their theoretical importance, their practical use remains challenging: $α$-stable probability densities generally do not have closed-form expressions, and numerical evaluation and random variate generation can be difficult, especially in the multivariate setting. This paper presents AUB-HTP, a Python package for numerical computation and simulation of $α$-stable distributions. The package provides scalar density evaluation using several complementary methods, including Zolotarev-type integral representations, series formulas, and numerical inversion of characteristic functions. It also provides random variate generation for scalar and multivariate $α$-stable distributions, with support for flexible spectral measures through LePage series representations. Numerical experiments demonstrate that AUB-HTP improves the accuracy, stability, and parameter coverage of existing tools for scalar density computation, while adding new capabilities for multivariate simulation. The package is designed to support reproducible computational work involving heavy-tailed models across a broad range of scientific applications.

Problem

• alpha-stable distributions
• heavy-tailed distributions
• density evaluation
• random number generation
• multivariate simulation

Innovation

• alpha-stable distributions
• numerical evaluation
• random variate generation
• multivariate simulation
• heavy-tailed models