This work focuses on the temporal average of the backward Euler--Maruyama (BEM) method, which is used to approximate the ergodic limit of stochastic ordinary differential equations with super-linearly growing drift coefficients. We give the central limit theorem (CLT) of the temporal average, which characterizes the asymptotics in distribution of the temporal average. When the deviation order is smaller than the optimal strong order, we directly derive the CLT of the temporal average through that of original equations and the uniform strong order of the BEM method. For the case that the deviation order equals to the optimal strong order, the CLT is established via the Poisson equation associated with the generator of original equations. Numerical experiments are performed to illustrate the theoretical results.

翻译：本文聚焦于倒向欧拉-马鲁亚马(BEM)法的时均，该方法用于逼近漂移系数超线性增长随机常微分方程的遍历极限。我们给出了时均的中心极限定理(CLT)，该定理刻画了时均的渐近分布特性。当偏差阶小于最优强阶时，我们通过原始方程的CLT及BEM法的均匀强阶直接推导出时均的CLT。对于偏差阶等于最优强阶的情形，则借助与原始方程生成元相关联的泊松方程建立CLT。通过数值实验验证了理论结果。