  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.
