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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
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