Sequential stopping rules malfunctioning

Abstract Sequential stopping rules provide a dynamic approach to determining when to conclude a process, basing the decision on real-time data instead of adhering to a fixed termination point. In the present study, while revisiting their well-known challenges, we uncover a wide range of previously unrecognized pitfalls arising when applying the established asymptotic theory in the practical non-asymptotic regime. Through illustrative discussions on deliberately straightforward examples, we highlight various potential risks associated with the behavior of empirical variances. By revealing these issues, we aim to emphasize the need for proactive planning and critical assessment to ensure the validity and reliability of sequential stopping rules, while also inspiring further research on approaches tailored to advanced modern frameworks involving stochastic simulations.