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 Π η of the EM scheme as a statistic of π η , and use Itô's formula and the Stein's method developed in Fang et al. ( 2019) to prove a central limit theorem of Π 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 (Π η (.) − π(.)) into a martingale difference series sum H η and a negligible remainder R η , we apply the SNCMD result for martingale in Fan et al. (2019) to H η and prove that R η is exponentially negligible. The concentration inequalities for estimating R η have independent interest, their proofs' strategy is explained by an Itô's formula and an exponential martingale. 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 (1997);Jing et al. (2003).
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.
In this paper, we obtain a Bernstein-type concentration inequality and McDiarmid’s inequality under upper probabilities for exponential independent random variables. Compared with the classical result, our inequalities are investigated under a family of probability measures, rather than one probability measure. As applications, the convergence rates of the law of large numbers and the Marcinkiewicz–Zygmund-type law of large numbers about the random variables in upper expectation spaces are obtained.
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