  The paper considers an Euler discretization based numerical scheme for
approximating functionals of invariant distribution of an ergodic diffusion.
Convergence of the numerical scheme is shown for suitably chosen discretization
step, and a thorough error analysis is conducted by proving central limit
theorem and moderate deviation principle for the error term. The paper is a
first step in understanding efficiency of discretization based numerical
schemes for estimating invariant distributions, which is comparatively much
less studied than the schemes used for generating approximate trajectories of
diffusions over finite time intervals. The potential applications of these
results also extend to other areas including mathematical physics, parameter
inference of ergodic diffusions and analysis of multiscale dynamical systems
with averaging.

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