  We consider a stochastic differential equation and its Euler-Maruyama (EM)
scheme, under some appropriate conditions, they both admit a unique invariant
measure, denoted by $\pi$ and $\pi_\eta$ respectively ($\eta$ is the step size
of the EM scheme). We construct an empirical measure $\Pi_\eta$ of the EM
scheme as a statistic of $\pi_\eta$, and use Stein's method developed in
\citet{FSX19} to prove a central limit theorem of $\Pi_\eta$. The proof of the
self-normalized Cram\'er-type moderate deviation (SNCMD) is based on a standard
decomposition on Markov chain, splitting $\eta^{-1/2}(\Pi_\eta(.)-\pi(.))$ into
a martingale difference series sum $\mcl H_\eta$ and a negligible remainder
$\mcl R_\eta$. We handle $\mcl H_\eta$ by the time-change technique for
martingale, while prove that $\mcl R_\eta$ is exponentially negligible by
concentration inequalities, which have their independent interest. Moreover, we
show that SNCMD holds for $x = o(\eta^{-1/6})$, which has the same order as
that of the classical result in \citet{shao1999cramer,JSW03}.
