论文标题
中央限制定理和自称cramér-type适度的欧拉 - 玛丽亚山计划
Central limit theorem and Self-normalized Cramér-type moderate deviation for Euler-Maruyama Scheme
论文作者
论文摘要
我们考虑一个随机微分方程及其Euler-Maruyama（EM）方案，在某些适当的条件下，他们都承认了一个独特的不变度度量，分别用$π$和$π_η$表示（$η$是EM方案的步骤大小）。我们构建了EM方案的经验度量$π_η$作为$π_η$的统计量，并使用在\ citet {fsx19}中开发的Stein方法来证明$π_η$的中心限制定理。自称cramér-type中度偏差（SNCMD）的证明是基于对马尔可夫链上的标准分解，将$η^{ - 1/2}（π_η（。我们按时间变化技术来处理$ \ mclh_η$，而Martingale的时间变化技术则证明$ \ mclr_η$被指数化，因为它具有其独立利益的集中度不平等。此外，我们表明SNCMD以$ x = o（η^{ - 1/6}）$持有，该$的顺序与\ citet {shao199999cramer，jsw03}中的经典结果的顺序相同。
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 Stein's method developed in \citet{FSX19} 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 $\mcl H_η$ and a negligible remainder $\mcl R_η$. We handle $\mcl H_η$ by the time-change technique for martingale, while prove that $\mcl R_η$ 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 \citet{shao1999cramer,JSW03}.
