analyze stopping times

This paper addresses the fragmented research landscape on Monte Carlo sequential stopping rules—characterized by inconsistent assumptions and a lack of systematic unification—by establishing, for the first time, a unified analytical framework encompassing both standard and moderately generalized Monte Carlo methods. Through a comprehensive review of over one hundred studies, it clarifies theoretical connections and practical distinctions among stopping rules under varying assumptions, and proposes a taxonomy covering convergence criteria, error control, and algorithmic design. Leveraging sequential analysis theory and integrating the law of large numbers with the central limit theorem, the work introduces a dynamic sample-size determination mechanism and an adaptive simulation strategy. The resulting framework significantly improves computational efficiency and enables precise, controllable error bounds. It provides a reusable theoretical foundation and practical guidelines for statistical inference, advanced Monte Carlo techniques—including MCMC and importance sampling—and complex simulation applications.

本文针对固定宽度顺序停止规则在无限方差或长程依赖情况下的失效问题，提出了一种基于联合泛函极限定理的方法，并引入了序列子抽样过程来解决。

本文针对强凸随机优化中固定时间保证与实际自适应停止决策之间的不匹配问题，通过构建可观察的、轨迹自适应的置信序列来解决，允许在保证精度的同时自适应停止SGD。

This work addresses the problem of optimizing finite-sample stopping times under fixed confidence in active hypothesis testing, with a focus on how hypothesis elimination influences decision efficiency. The authors propose a Track-and-Stop algorithm enhanced with a hypothesis elimination mechanism that dynamically prunes competing hypotheses and reallocates sensing resources to accelerate identification of the true hypothesis. They innovatively quantify, within a finite-sample analysis, the improvement hypothesis elimination brings to non-dominant terms of the stopping time bound and introduce a tunable aggressiveness parameter to balance elimination speed against confidence guarantees. Leveraging techniques from sequential hypothesis testing, adaptive sampling, and concentration inequalities, they derive a non-asymptotic upper bound on the expected stopping time and validate the theoretical predictions through experiments on synthetic Gaussian data.

This study addresses the Kiefer–Weiss problem in a nonparametric setting, aiming to minimize a weighted sum of error probabilities in binary sequential hypothesis testing under a constraint on the worst-case expected sample size. By reformulating the problem as an optimal stopping problem, the authors introduce finitely randomized strategies and take their limit to derive, for the first time, a complete solution. The resulting optimal stopping rule is characterized by a two-dimensional statistic comprising the likelihood ratio and the remaining allowable sample size. A key innovation lies in uncovering a novel mechanism—dynamic sample-size adjustment via randomization—that enhances detection performance. Practical approximate strategies are also proposed. The methodology is applicable to both Bernoulli success probability testing and normal mean shift detection, with numerical experiments confirming its efficacy.

We give a necessary and sufficient condition for the existence of power-one sequential tests in an i.i.d. composite testing problem. A level-\(\alpha\) test with power one against every alternative exists if and only if the alternatives are separated from the null by a countable family of finite-block events. We provide other equivalent conditions using randomized fixed-sample tests, bounded finite-block scores, e-processes, reduced-filtration test supermartingales, and a countable cover whose finite-block weak-$*$ closed convex hulls are positively separated in total variation. As a bonus, the constructive proof yields tests have pointwise expected sample size \(O_Q(\log(1/\alpha))\). Exactly the same conditions also characterize i.i.d.\ change detectability under optional-horizon average-run-length control: for every \(\eta>0\), they are equivalent to an alarm family \((T_\gamma)_{\gamma\ge1}\) satisfying \(\Prob_{P^\infty}(T_\gamma\le\sigma)\le \E_{P^\infty}\sigma/\gamma\) for every null law and every stopping time \(\sigma\). In fact, when these conditions hold, we can construct a single e-detector such that every null-law average run length lies between \(\gamma\) and \((1+\eta)\gamma+1\), and having robust Lorden delay \(O_Q(\log\gamma)\).

本文研究了基于鞅理论的多项式不变量合成方法，用于概率转移系统，并提出了一种可处理的方法来验证这些多项式确实是不变量，允许采样分布具有无界支持。

This work addresses the challenge of controlling multiple testing errors in online hypothesis testing, where hypotheses arrive dynamically and evidence becomes observable at arbitrary times. The authors propose a dynamic e-closure method that integrates e-values, dynamic closure principles, and cross-time consistency constraints to establish the first theoretical framework capable of effectively controlling the supremum false discovery rate (SupFDR) under simultaneous stopping and ensuring the persistence of rejection sets. Key contributions include establishing universal guarantees under a canonical normalized loss process, revealing structural limitations inherent to pointwise merging procedures, and constructing a globally shared control procedure via projection merging, alongside a counterexample demonstrating the failure of consistency under certain conditions.

研究解决了折扣最小二乘估计器在非平稳问题中的自归一化集中不等式的错误，并通过修正方法保证了其在固定时间的有效性。

This study addresses the excessive conservativeness of universal e-value thresholds under structured null distributions by proposing a calibration method based on non-decreasing and Lipschitz density models. Theoretically, we derive asymptotically optimal Lipschitz-calibrated thresholds by leveraging Markov inequality constraints, minimax theory, and conditional e-value construction techniques. Practically, this approach is extended to false discovery rate (FDR) control and sequential analysis settings. The proposed method substantially increases the number of discoveries in multiple testing scenarios. Furthermore, it demonstrates a favorable trade-off between statistical power and threshold sharpness within sequential procedures. Overall, this work provides a more efficient solution for statistical inference under structured null hypotheses.