Central limit theorem and self-normalized Cramér-type moderate deviation for Euler-Maruyama scheme

Abstract

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