Cramér type moderate deviations for self-normalized ψ-mixing sequences

Fan, Xiequan

doi:10.1016/j.jmaa.2020.123902

Cited by 4 publications

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References 26 publications

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“…The other is that our result provides a new example for SNCMD for dependent time series, and also a new example that applies Stein's method to prove SNCMD, see Chen, Fang and Shao [7], Shao, Zhang and Zhang [37]. Note that there are not many results for SNCMD for dependent time series, see Chen et al [8], Fan [15], Fan et al [16,17], Fang, Luo and Shao [18], Jing, Wang and Zhou [21], Shao and Zhou [38] and the references therein.…”

Section: Introductionmentioning

confidence: 80%

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

Tan

2022

Bernoulli

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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 π η respectively (η is the step size of the EM scheme). We construct an empirical measure η of the EM scheme as a statistic of π η , 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 η . The proof of the self-normalized Cramér-type moderate deviation (SNCMD) is based on a standard decomposition on Markov chain, splitting η −1/2 ( η (.) − π(.)) into a martingale difference series sum H η and a negligible remainder R η . We handle H η by the time-change technique for martingale, while prove that R η 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 (J.

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