Stopping Rules for Monte Carlo Methods: A Review

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.