## Central limit theorem and self-normalized Cramér-type moderate deviation for Euler-Maruyama scheme.(English)Zbl 1489.60039

Summary: 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 [X. Fang et al., Probab. Theory Relat. Fields 174, No. 3–4, 945–979 (2019; Zbl 1479.60003)] to prove a central limit theorem of $$\Pi_{\eta}$$. The proof of the self-normalized Cramér-type moderate deviation (SNCMD) is based on a standard decomposition on Markov chains, splitting $$\eta^{-1/2}(\Pi_{\eta}(.)-\pi (.))$$ into a martingale difference series sum $$\mathcal{H}_{\eta}$$ and a negligible remainder $$\mathcal{R}_{\eta}$$. We handle $${\mathcal{H}_{\eta}}$$ by the time-change technique for martingales, while prove that $$\mathcal{R}_{\eta}$$ 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 [Q.-M. Shao, J. Theor. Probab. 12, No. 2, 385–398 (1999; Zbl 0927.60045); B.-Y. Jing et al., Ann. Probab. 31, No. 4, 2167–2215 (2003; Zbl 1051.60031)].

### MSC:

 60F05 Central limit and other weak theorems 60F10 Large deviations 62E20 Asymptotic distribution theory in statistics 62F03 Parametric hypothesis testing 60G50 Sums of independent random variables; random walks 65C30 Numerical solutions to stochastic differential and integral equations

### Citations:

Zbl 1479.60003; Zbl 0927.60045; Zbl 1051.60031
Full Text:

### References:

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