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arxiv: 2606.28530 · v1 · pith:FE7HTFXZnew · submitted 2026-06-26 · 💻 cs.MS

Density Functions and Random Number Generators of α-Stable Distributions

Wael Tabbara , Sharafeddine Sharafeddine , Ahmad El-Hajj , Jihad Fahs , Ibrahim Abou-Faycal This is my paper

Pith reviewed 2026-06-30 01:00 UTC · model grok-4.3

classification 💻 cs.MS
keywords alpha-stable distributionsdensity evaluationrandom variate generationPython packageheavy-tailed modelsLePage seriesnumerical methodsmultivariate simulation
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The pith

AUB-HTP package improves accuracy and stability of alpha-stable density computation while enabling multivariate simulation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a Python package called AUB-HTP for computing probability densities and generating random samples from alpha-stable distributions. These distributions lack closed-form expressions, making numerical methods necessary for their use in modeling heavy-tailed data from fields like finance and physics. The package implements several complementary techniques for scalar densities and uses LePage series for both scalar and multivariate generation. Numerical tests indicate that it handles a wider range of parameters more reliably than previous software. This supports more practical application of generalized central limit theorem models in empirical work.

Core claim

AUB-HTP implements scalar density evaluation via Zolotarev-type integrals, series formulas, and characteristic function inversion, together with LePage series representations for random variate generation in scalar and multivariate settings, and experiments show gains in accuracy, stability, and parameter coverage over existing tools along with new multivariate capabilities.

What carries the argument

The AUB-HTP Python package, using complementary numerical methods including Zolotarev-type integral representations, series formulas, numerical inversion of characteristic functions for densities, and LePage series for simulation.

If this is right

  • Scalar density computations become feasible and accurate for parameter values where prior tools were unstable or inaccurate.
  • Multivariate alpha-stable random variates can now be generated for applications requiring joint heavy-tailed modeling.
  • Reproducible simulations involving sums of heavy-tailed sources are facilitated across engineering, finance, and network science.
  • Users gain flexibility in choosing spectral measures for the stable distributions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Integration with existing statistical libraries could allow seamless use in larger data analysis pipelines.
  • Extension to other related distributions like tempered stable laws might follow similar implementation strategies.
  • Validation against real-world datasets from physics or finance would further demonstrate practical utility.

Load-bearing premise

The numerical implementations are free of coding errors and the chosen test cases fairly represent the parameter regimes where existing tools fail, without post-hoc selection that favors the new package.

What would settle it

Finding a parameter combination where an existing density tool produces a result closer to a high-precision reference than AUB-HTP does, or where multivariate samples from AUB-HTP fail basic consistency checks like marginal stability.

Figures

Figures reproduced from arXiv: 2606.28530 by Ahmad El-Hajj, Ibrahim Abou-Faycal, Jihad Fahs, Sharafeddine Sharafeddine, Wael Tabbara.

Figure 1
Figure 1. Figure 1: shows agreement with the SciPy reference. Results for the remaining error metrics, parameter ranges, and tail evaluations, together with the complete benchmark data, are available in the Github repository3 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Visual comparison between AUB-HTP PDF generator and scipy.stats.levy stable.pdf. The plots zoom into regions where the SciPy reference becomes numerically zero or behaves irregularly. These regions explain why some relative difference values become large even when our generated density remains smooth. The heatmaps are useful for locating sensitive parameter regions, but the diagnostic plots were needed to … view at source ↗
Figure 3
Figure 3. Figure 3: Median runtime speedup relative to scipy.stats.levy stable.pdf. The largest gains occur where the series expansions are selected. The speedup plot in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Truncated histogram of α-stable distributions, each containing 5 × 105 points. An example using the type 0 parameterization is presented in [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Truncated histogram of α-stable distributions, each containing 5 × 105 points. 5.2 Multivariate random number generator Isotropic α-stable random vector: An isotropic α-stable random vector with characteristic function ϕX(t) = e −γ α∥t∥ α+it T µ 0 , can be generated using the AUB-HTP package as follows: 1 import aub_htp as ht 2 from aub_htp.random import IsotropicSampler 3 alpha = 0.5 4 gamma = 1 5 shift= … view at source ↗
Figure 6
Figure 6. Figure 6: Scatter plots each containing 5 × 105 points of isotropic α-stable distributions. The plots have been truncated to a range of 400 points within the shift vector. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: shows the scatter plots of the elliptical α-stable random vector, where we use the following shape matrices Σ1 =  2 0.8 0.8 1.5  Σ2 =  2 −1 −1 2  . (a) α = 0.5, Σ1, µ 0T = (0, 0) (b) α = 1, Σ1, µ 0T = (1000, 1000) (c) α = 1.1, Σ2, µ 0T = −(2000, 2000) [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Scatter plots each containing 5 × 105 points of α-stable distributions with a discrete spectral measure. The plots have been truncated to a range of 400 points within the shift vector. Mixed spectral measure: For a mixed spectral measure, let Λ1, · · · ,ΛL be L spectral measures with asso￾ciated weights w1, · · · , wL, and consider their mixture Λmix = PL i=1 wiΛi . To sample according to this spectral mea… view at source ↗
Figure 9
Figure 9. Figure 9: Scatter plots showing a custom spectral measure and different mixtures of spectral measures. The plots have been truncated to a range of 400 points within the shift vector. We highlight that the mass of the sphere and the dimension should be specified when implementing the sub-class. The resulting scatter plot is shown in Figure 9b. It is important to emphasize that when α ≥ 1, the custom spectral measure … view at source ↗
read the original abstract

Heavy-tailed distributions are increasingly found to better fit empirical data in engineering, finance, physics, network science, and related fields. Among them, $\alpha$-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: $\alpha$-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 $\alpha$-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 $\alpha$-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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript presents AUB-HTP, a Python package implementing multiple methods (Zolotarev-type integrals, series expansions, characteristic-function inversion) for scalar α-stable density evaluation and LePage-series representations for scalar and multivariate random-variate generation. It asserts that numerical experiments demonstrate superior accuracy, stability, and parameter coverage relative to existing tools, together with new multivariate simulation capabilities.

Significance. If the implementations prove correct and the performance claims are reproducible, the package would supply a missing, open-source resource for reliable α-stable modeling in heavy-tailed applications across finance, physics, and network science, especially for the multivariate case where few alternatives exist.

major comments (2)
  1. [Numerical Experiments] Numerical Experiments section (and abstract): the central claim of improved accuracy, stability, and coverage rests on unspecified test cases, error metrics, baselines, and parameter regimes; no quantitative tables, figures, or code are referenced, rendering the claim unverifiable.
  2. [Methods and Numerical Experiments] Implementation and validation: no cross-checks are reported against analytically known special cases (Gaussian limit α=2, Cauchy α=1, Lévy α=1/2) or against independent libraries, leaving open the possibility that reported gains arise from implementation artifacts rather than methodological improvement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which identify key areas where additional detail will strengthen the verifiability of our claims. We address each major comment below and commit to the indicated revisions.

read point-by-point responses

  1. Referee: [Numerical Experiments] Numerical Experiments section (and abstract): the central claim of improved accuracy, stability, and coverage rests on unspecified test cases, error metrics, baselines, and parameter regimes; no quantitative tables, figures, or code are referenced, rendering the claim unverifiable.

    Authors: We agree that the current manuscript does not provide sufficient detail on the test cases, error metrics, baselines, and parameter regimes underlying the accuracy and stability claims. We will revise the Numerical Experiments section (and update the abstract accordingly) to include explicit quantitative tables of errors, figures showing performance across parameter ranges, named baselines (e.g., comparisons to scipy.stats and other libraries), and references to the accompanying code and data for full reproducibility. revision: yes

  2. Referee: [Methods and Numerical Experiments] Implementation and validation: no cross-checks are reported against analytically known special cases (Gaussian limit α=2, Cauchy α=1, Lévy α=1/2) or against independent libraries, leaving open the possibility that reported gains arise from implementation artifacts rather than methodological improvement.

    Authors: We concur that validation against analytically tractable special cases and independent implementations is necessary to substantiate the reported gains. We will add a new subsection reporting direct comparisons for α=2 (Gaussian), α=1 (Cauchy), and α=1/2 (Lévy) against closed-form expressions, as well as numerical cross-checks against at least one established external library for both density evaluation and variate generation. revision: yes

Circularity Check

0 steps flagged

No circularity: implementation of established methods

full rationale

The paper presents AUB-HTP, a Python package implementing known numerical methods (Zolotarev integrals, series expansions, characteristic function inversion, LePage representations) for α-stable distributions. No derivation, parameter fitting, or prediction is claimed that reduces by construction to its own inputs. Claims rest on numerical experiments comparing to existing tools, but no self-referential reduction or load-bearing self-citation chain is present. This matches the default expectation of no significant circularity for implementation-focused work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on the assumption that the chosen numerical algorithms are correctly coded and that the reported experiments accurately reflect real-world performance across parameter space; no free parameters, axioms, or invented entities are introduced.

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discussion (0)

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Reference graph

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