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The paper considers an Euler discretization based numerical scheme forapproximating functionals of invariant distribution of an ergodic diffusion.Convergence of the numerical scheme is shown for suitably chosen discretizationstep, and a thorough error analysis is conducted by proving central limittheorem and moderate deviation principle for the error term. The paper is afirst step in understanding efficiency of discretization based numericalschemes for estimating invariant distributions, which is comparatively muchless studied than the schemes used for generating approximate trajectories ofdiffusions over finite time intervals. The potential applications of theseresults also extend to other areas including mathematical physics, parameterinference of ergodic diffusions and analysis of multiscale dynamical systemswith averaging.

Inhomogeneous functionals and approximations of invariant distributions  of ergodic diffusions: Error analysis through central limit theorem and  moderate deviation asymptotics