May 2022 Central limit theorem and self-normalized Cramér-type moderate deviation for Euler-Maruyama scheme
Jianya Lu, Yuzhen Tan, Lihu Xu
Author Affiliations +
Bernoulli 28(2): 937-964 (May 2022). DOI: 10.3150/21-BEJ1372


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 (Probab. Theory Related Fields 174 (2019) 945–979) 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 η1/2(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(η1/6), 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).

Funding Statement

LX is supported in part by NSFC grant 12071499, Macao S.A.R grant FDCT 0090/2019/A2 and University of Macau grant MYRG2018-00133-FST.


We would like to gratefully thank Professors Fuqing Gao and Feng-Yu Wang for very helpful discussions. We also thank two anonymous referees and the AE for their valuable comments which have improved the manuscript considerably.

Lihu Xu as the corresponding author.


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Jianya Lu. Yuzhen Tan. Lihu Xu. "Central limit theorem and self-normalized Cramér-type moderate deviation for Euler-Maruyama scheme." Bernoulli 28 (2) 937 - 964, May 2022.


Received: 1 December 2020; Revised: 1 May 2021; Published: May 2022
First available in Project Euclid: 3 March 2022

MathSciNet: MR4388925
zbMATH: 1489.60039
Digital Object Identifier: 10.3150/21-BEJ1372

Keywords: central limit theorem , Euler-Maruyama scheme , self-normalized Cramér-type moderate deviation , Stein’s method , Stochastic differential equation

Rights: Copyright © 2022 ISI/BS


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Vol.28 • No. 2 • May 2022
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