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