asymptotic analysis

The empirical statistical behavior of sample skewness and kurtosis under small-sample regimes remains poorly understood. Method: Leveraging asymptotic analysis, heavy-tailed distribution modeling, and experiments on both synthetic and real-world datasets, we derive rigorous theoretical bounds and examine moment estimation properties. Contribution/Results: We establish, for the first time, a strict asymptotic lower bound on sample kurtosis as a function of sample size and skewness. Extending the Taylor power law to higher-order moments, we reveal that the skewness–kurtosis 4/3 scaling law arises intrinsically from the asymptotic structure of heavy-tailed distributions. This scaling is robust only for heavy-tailed data and sufficiently large samples (n ≳ 100); under small samples, kurtosis exhibits a pronounced negative bias with a nontrivial lower bound, rendering classical moment estimators severely biased. Our results provide a theoretically grounded criterion for the validity domain of moment-based estimation and correct the misconception of universal power-law applicability.

This paper addresses the problem of assessing clustering stability for one-dimensional data under extreme-point trimming. We propose a probabilistic analysis method based on iterative extremal-point pruning. Its core contribution is the introduction of the “diameter shrinkage ratio” — a parameter-free, density-estimation-free metric for quantifying stability — derived via analytical modeling (including finite-sample corrections) for uniform and Gaussian distributions, yielding closed-form expected shrinkage curves. A simple decision rule enables automatic distribution identification and cluster stability verification. The method requires no hyperparameter tuning and significantly outperforms classical likelihood-ratio tests in small-sample and noisy settings, while maintaining asymptotic consistency. It integrates seamlessly into clustering pipelines such as DBSCAN. Experiments demonstrate superior classification accuracy, establishing a robust, interpretable, and principled validation paradigm for one-dimensional clustering.

This study addresses the risk of misclassification in process capability assessment under limited sample sizes when using fixed thresholds such as Cpk ≥ 1.33, due to the stochastic nature of estimators near decision boundaries. The work establishes, for the first time, a probabilistic framework for Cpk-based threshold decisions and employs local asymptotic theory to demonstrate that when the true process capability deviates from the threshold by an order of O(n⁻¹/²), the decision probability converges to a non-degenerate limit governed by the signal-to-noise ratio. Through Monte Carlo simulations, resampling techniques, and empirical analysis across 880 manufacturing dimensions, the study quantifies substantial decision instability around commonly used thresholds, thereby providing both theoretical justification and practical tools for evaluating borderline release risks in engineering practice.

This paper addresses precise large-deviation estimation of tail probabilities for threshold exceedances in triangular arrays of conditionally dependent latent-factor models, focusing on how tail geometry and latent dependence shape the leading-order asymptotics. For three tail classes—Gaussian/exponential-power, regularly varying, and bounded support—we derive refined decay rates with prefactors of logarithmic polynomials, power laws, and $n^{-3/2}$, respectively. Methodologically, we develop a second-order characterization framework integrating localization, curvature analysis, and tilt identification. Combining Laplace–Olver asymptotics, conditional Bahadur–Rao approximations, and endpoint analysis, we establish a conditional Gibbs principle in total variation distance, revealing asymptotic independence and tilted loss-distribution structure under large-scale default events. The results yield second-order approximations for Value-at-Risk (VaR) and Expected Shortfall (ES), and explicitly characterize the entry criterion into the large-deviation regime.

This paper investigates the convergence rate of the halfspace depth empirical estimator, establishing for the first time a quantitative relationship between this rate and the tail index of the underlying distribution (e.g., Weibull- or Pareto-type tails). Methodologically, it integrates weighted empirical process theory, extreme value analysis, and multivariate nonparametric statistics to derive universal upper bounds on the convergence rate, explicitly characterizing the joint influence of sample size and tail parameters. A key contribution is a novel, depth-based framework for tail-type discrimination—applicable uniformly to both light- and heavy-tailed multivariate distributions—leveraging the asymptotic decay rate of depth values. Extensive simulations and real-data experiments demonstrate that the proposed method substantially improves accuracy in multivariate tail identification, offering a new tool for modeling tail behavior in high-dimensional distributions.

Existing extreme-value models struggle to flexibly characterize both marginal tail behavior and joint dependence structures of bivariate threshold exceedances under asymptotic independence. This work proposes a novel bivariate subasymptotic parametric model that, while ensuring convergence of the margins to the generalized Pareto distribution, naturally captures the evolution of extremal dependence with varying thresholds through its scale parameters. The model encompasses the standard multivariate generalized Pareto distribution as a limiting special case and accommodates a broad spectrum of tail dependence patterns. Inference is carried out via a likelihood-free neural Bayesian approach with tailored priors, enabling direct computation and interpretation of failure probabilities. Extensive simulations and an analysis of Belgian rainfall extremes demonstrate the model’s flexibility and the effectiveness of the proposed inference framework.

This work addresses the lack of a unified computational framework for analyzing non-ergodicity, modeling heavy-tailed dynamics, and studying decision-making under uncertainty in stochastic processes. To this end, we introduce an open-source Python library that, for the first time, integrates non-ergodicity diagnostics, simulation of heavy-tailed processes—such as multiplicative Lévy growth and memory-dependent mean-reverting dynamics—and agent-based experimentation within a single platform. Built upon the scientific Python ecosystem (NumPy/SciPy), the library supports end-to-end workflows including stochastic process definition, simulation, parameter inference, and partial solution of stochastic differential equations. Through several reproducible examples—ranging from heavy-tailed ensemble diffusion to pre-asymptotic fluctuation analysis—it substantially reduces boilerplate code and enhances both reproducibility and development efficiency in the study of time-averaged behaviors of complex stochastic systems.

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.

This work proposes a universal framework for identifying phase transitions without requiring order parameters or prior knowledge of the underlying model. Building on the hypothesis that infinitesimal parameter perturbations break statistical indistinguishability in the thermodynamic limit, the study redefines phase transitions as abrupt changes in distributional distinguishability. This perspective enables a model-agnostic, training-free detection method implemented via a distribution-free two-sample runs test. The approach unifies conventional criteria—including the Binder cumulant—and accurately locates the critical point in the two-dimensional Ising model, thereby demonstrating both its validity and broad applicability across different systems.