Central limit theorem and self-normalized Cramér-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 unique invariant measure, denoted by π πη respectively (η is the step size of EM scheme). construct an empirical measure Πη scheme as statistic πη, use Stein’s method developed in Fang, Shao Xu (Probab. Theory Related Fields 174 (2019) 945–979) to prove central limit theorem Πη. The proof self-normalized Cramér-type moderate deviation (SNCMD) based on standard decomposition Markov chain, splitting η−1/2(Πη(.)−π(.)) into martingale difference series sum Hη negligible remainder Rη. handle time-change technique for martingale, while that Rη exponentially concentration inequalities, which have their independent interest. Moreover, we show SNCMD holds x=o(η−1/6), has same order classical result (J. Theoret. Probab. 12 (1999) 385–398), Jing, Wang (Ann. 31 (2003) 2167–2215).