analyze asymptotic concentration

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.

This paper identifies the systematic failure of classical statistical methods—including mean-based inference, principal component analysis (PCA), and asymptotic normality assumptions—under heavy-tailed distributions, particularly in medium-sample-size (medium-*n*) real-world settings where they are routinely misapplied. Methodologically, it challenges the uncritical adoption of Gaussian and stable-distribution assumptions and introduces the “Median Law” theoretical framework, which formalizes fundamental limitations under heavy tails: unreliable sample means, distorted empirical distributions, and degenerate principal components. The approach integrates extreme value theory, generalized stable distribution modeling, robust parametric estimation, and pre-asymptotic analysis. Empirical validation draws on counterexamples from finance, economics, and psychology, supplemented by cross-disciplinary case studies. The core contribution is a foundational rethinking of uncertainty quantification and causal inference: it demonstrates that many canonical “cognitive biases” are, in fact, rational inferences under heavy-tailed probability structures—thereby advocating a paradigm shift in statistical practice from idealized asymptotics to empirically grounded probabilistic modeling.

This paper addresses asymptotic inference for a single large network sample generated by strategic interaction and homophily among agents in large-scale static and dynamic network formation models. To overcome the challenge of verifying conventional central limit theorems (CLTs) under network moment dependence, we adapt the exponential stabilization condition from stochastic geometry to network analysis—augmented by branching process theory—to derive verifiable primitive sufficient conditions. The resulting CLT framework requires no repeated sampling and applies directly to a single large network. It substantially broadens the theoretical foundation for network parameter estimation and hypothesis testing, and provides the first asymptotic normality guarantee for strategic network models with explicit, quantifiable regularity conditions.

This work establishes Rosenthal- and Bernstein-type concentration inequalities for additive functionals of geometrically ergodic Markov chains, explicitly characterizing the dependence of deviation bounds on mixing time. Methodologically, it pioneers the extension of the classical Rosenthal inequality to the Markov-dependent setting via a novel analytical framework based on Poisson equation decomposition, which precisely links mixing constants, martingale Rosenthal constants, and deviation bounds. Integrating martingale techniques, geometric ergodicity analysis, and quantitative mixing time estimation, the approach yields computable, explicit, and tight upper bounds—significantly improving the polynomial dependence on mixing time present in prior results. The derived inequalities provide a rigorous theoretical foundation for error control in MCMC algorithms, sequential Monte Carlo estimation, and large-sample inference for non-i.i.d. statistics.

High-probability analysis of learning algorithms involving light-tailed (e.g., sub-exponential, sub-Gaussian) but possibly unbounded random variables poses significant technical challenges due to the lack of uniform concentration tools across distribution families. Method: We propose a generic black-box reduction that systematically transforms high-probability analysis of any algorithm relying on light-tailed randomness into the corresponding analysis under bounded-variable assumptions, incurring only controllable logarithmic-factor overheads. Contribution/Results: This is the first unified framework handling diverse light-tailed distributions without ad hoc concentration inequalities—greatly simplifying theoretical analysis. As applications, we reconstruct a generalized Azuma’s inequality and derive tight high-probability convergence bounds for stochastic optimization algorithms under light-tailed noise, demonstrating both the method’s effectiveness and broad applicability.

Existing sequential mean testing methods struggle to characterize the higher-order asymptotic behavior of stopping times, particularly lacking second-order information-theoretic optimality analyses under bounded distributions. This work proposes a sequential test based on the KL_inf statistic, which not only achieves first-order asymptotic optimality but also establishes, for the first time, a central limit theorem for this statistic. Consequently, the stopping time—after appropriate centering and scaling by √log(1/α)—converges in distribution to a Gaussian limit. This result provides a refined second-order characterization of sequential test performance under bounded distributions, with theoretical proof showing convergence to a normal distribution having an explicit variance. Numerical experiments corroborate the accuracy of this second-order asymptotic analysis.

This work investigates the concentration of iteration errors in stochastic approximation algorithms driven by heavy-tailed Markov noise, covering both expansive and non-expansive operator settings. Under a framework involving a finite-state Markov component and martingale difference noise, the authors construct a novel Lyapunov function via the moment-generating function of the solution to the Poisson equation, complemented by auxiliary projection and black-box truncation techniques to reduce unbounded noise to a bounded setting. The study provides the first systematic characterization of the fine structure of error tails: under bounded noise, tails can be sub-Gaussian, sub-Weibull, or intermediate between Pareto and Weibull; under unbounded noise, if the operator is almost surely non-expansive, the error tail is at most three times heavier than that of the noise, whereas if the operator is expansive with positive probability, significantly heavier tails may arise, with sharp worst-case examples demonstrating the tightness of these bounds.

论文通过Bernstein函数和Hausdorff序列模型探讨词汇增长理论，解决了词汇类型预期数量建模问题，并分析了该理论在不同随机过程下的局限性。

This study addresses the unresolved non-asymptotic sample complexity of single-trajectory least squares estimation for linear systems under heavy-tailed noise. For exponentially stable systems, this work proposes a unified analytical framework that accommodates both sub-Gaussian and sub-exponential noise regimes, subject to persistence of excitation, bounded noise covariance, and finite moment conditions. The primary contribution lies in establishing a non-asymptotic error bound of $\widetilde{O}(r^{1/2}T^{-1/2+1/p})$, which is shown to be independent of the model order. This result significantly improves system identification performance in the presence of heavy-tailed noise, offering tighter theoretical guarantees than existing approaches.

研究通过路径空间信息流，使用变分恒等式方法解决非负鞅在随机时间点的精确控制问题，并分析了多种几何形态下的舍弃松弛。