We establish a practical and easy-to-implement sequential stopping rule for the martingale central limit theorem, focusing on Monte Carlo methods for estimating the mean of a non-iid sequence of martingale difference type. Starting with an impractical scheme based on the standard martingale central limit theorem, we progressively address its limitations from implementation perspectives in the non-asymptotic regime. Along the way, we compare the proposed schemes with their counterparts in the asymptotic regime. The developed framework has potential applications in various domains, including stochastic gradient descent methods. Numerical results are provided to demonstrate the effectiveness of the developed stopping rules in terms of reliability and complexity.

翻译：本文针对鞅中心极限定理建立了一种实用且易于实现的序贯停止规则，重点关注用于估计非独立同分布鞅差型序列均值的蒙特卡洛方法。我们从基于标准鞅中心极限定理的不可行方案出发，逐步从非渐近区域中的实现角度解决其局限性。在此过程中，我们将所提出的方案与渐近区域中的对应方案进行比较。所开发的框架在多个领域具有潜在应用价值，包括随机梯度下降方法。数值结果展示了所开发停止规则在可靠性和计算复杂度方面的有效性。