Central limit theorem and Self-normalized Cram\'er-type moderate deviation for Euler-Maruyama Scheme

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}.