Extending Subsampling to Sequential Stopping

2026-09-01

AI Summary

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。

Abstract

Fixed-width sequential stopping rules terminate a stochastic simulation once an estimated confidence interval reaches a prescribed width. Classical fixed-width theory typically relies on a strongly consistent estimator of the asymptotic variance. This makes the normalized stopping time asymptotically deterministic, allowing fixed-sample-size limit theory to be transferred to the estimator at termination. This mechanism can fail when simulation output has infinite variance or long-range dependence. Although self-normalization and subsampling can yield asymptotically valid confidence intervals at a fixed sample size, the scaling process and the stopping time may retain nondegenerate randomness, so fixed-sample-size quantiles need not provide valid coverage at termination. In this paper, we develop a unified framework based on a joint functional limit theorem for the estimation process and a scaling process. We characterize the asymptotic behavior of both the stopping time and the self-normalized estimator evaluated at termination, thereby obtaining asymptotically valid sequential confidence intervals in classical finite-variance and infinite-variance settings. Moreover, we introduce a sequential subsampling procedure that consistently estimates the distribution relevant at the stopping time without directly estimating nuisance parameters in the limit distribution. The framework is verified for heavy-tailed moving-average processes, stochastic approximation, and an M/G/1 queue with heavy-tailed service times.

Problem

• sequential stopping
• infinite variance
• long-range dependence

Innovation

• sequential subsampling
• functional limit theorem
• asymptotically valid confidence intervals
• infinite-variance settings