In order to give quantitative estimates for approximating the ergodic limit, we investigate probabilistic limit behaviors of time-averaging estimators of numerical discretizations for a class of time-homogeneous Markov processes, by studying the corresponding strong law of large numbers and the central limit theorem. Verifiable general sufficient conditions are proposed to ensure these limit behaviors, which are related to the properties of strong mixing and strong convergence for numerical discretizations of Markov processes. Our results hold for test functionals with lower regularity compared with existing results, and the analysis does not require the existence of the Poisson equation associated with the underlying Markov process. Notably, our results are applicable to numerical discretizations for a large class of stochastic systems, including stochastic ordinary differential equations, infinite dimensional stochastic evolution equations, and stochastic functional differential equations.

翻译：为给出遍历极限逼近的定量估计，我们通过研究相应的强大数定律与中心极限定理，探讨一类时间齐次马尔可夫过程数值离散的时间平均估计量的概率极限行为。提出可验证的通用充分条件以确保这些极限行为，该条件与马尔可夫过程数值离散的强混合性与强收敛性质相关。与现有结果相比，我们的结论适用于正则性更低的测试泛函，且分析无需依赖底层马尔可夫过程伴随的泊松方程。值得注意的是，本结果适用于包含随机常微分方程、无穷维随机发展方程及随机泛函微分方程在内的大类随机系统的数值离散。