Stopping Rules for Monte Carlo Methods: A Review

TL;DR

This review surveys sequential stopping rules for standard and mildly generalized Monte Carlo methods, tracing developments from early asymptotic theory to modern non-asymptotic, moment-based, and distribution-specific approaches. It organizes the landscape around error tolerances, confidence levels, multivariate outputs, non-iid data, and single- vs two-stage procedures, and then details recent advances in asymptotic frameworks, Berry-Esseen refinements, and nonstandard contexts like quasi-Monte Carlo and Markovian settings. Key contributions include frameworks for asymptotic analysis, higher-order corrections, and a taxonomy of stopping rules with guaranteed error or coverage properties under various assumptions, along with quality assessment tools such as coverage profiles. The paper highlights practical challenges, emphasizes the value of two-stage designs when variances are unknown, and identifies opportunities to merge sequential stopping with variance reduction and learning-based methods for broader, more actionable deployments in Monte Carlo inference.

Abstract

Sequential analysis encompasses simulation theories and methods where the sample size is determined dynamically based on accumulating data. Since the conceptual inception, numerous sequential stopping rules have been introduced, and many more are currently being refined and developed. Those studies often appear fragmented and complex, each relying on different assumptions and conditions. This article aims to deliver a comprehensive and up-to-date review of recent developments on sequential stopping rules, intentionally emphasizing standard and moderately generalized Monte Carlo methods, which have historically served, and likely will continue to serve, as fundamental bases for both theoretical and practical developments in stopping rules for general statistical inference, advanced Monte Carlo techniques and their modern applications. Building upon over a hundred references, we explore the essential aspects of these methods, such as core assumptions, numerical algorithms, convergence properties, and practical trade-offs to guide further developments, particularly at the intersection of sequential stopping rules and related areas of research.