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英文文献：Quantitative Breuer-Major Theorems-定量Breuer-Major定理
英文文献作者：Ivan Nourdin,Giovanni Peccati,Mark Podolskij
英文文献摘要：
We consider sequences of random variables of the type $S_n= n^{-1/2} \sum_{k=1}^n f(X_k)$, $n\geq 1$, where $X=(X_k)_{k\in \Z}$ is a $d$-dimensional Gaussian process and $f: \R^d \rightarrow \R$ is a measurable function. It is known that, under certain conditions on $f$ and the covariance function $r$ of $X$, $S_n$ converges in distribution to a normal variable $S$. In the present paper we derive several explicit upper bounds for quantities of the type $|\E[h(S_n)] -\E[h(S)]|$, where $h$ is a sufficiently smooth test function. Our methods are based on Malliavin calculus, on interpolation techniques and on the Stein's method for normal approximation. The bounds deduced in our paper depend only on $\E[f^2(X_1)]$ and on simple infinite series involving the components of $r$. In particular, our results generalize and refine some classic CLTs by Breuer-Major, Giraitis-Surgailis and Arcones, concerning the normal approximation of partial sums associated with Gaussian-subordinated time-series.
我们考虑$S_n= n^{-1/2} \sum_{k=1}^n f(X_k)$， $n\geq 1$，其中$X=(X_k)_{k\in \Z}$是一个$d$-维高斯过程，$f: \R^d \右行\R$是一个可测量的函数。已知，在$f$和$X$的协方差函数$r$的一定条件下，$S_n$在分布上收敛于正态变量$S$。在本文中，我们推导了$|\E[h(S_n)] -\E[h(S)]|$这类量的几个显式上界，其中$h$是一个充分光滑的测试函数。我们的方法是基于Malliavin微积分、插值技术和Stein的正态逼近方法。本文推导出的界只取决于$\E[f^2(X_1)]和包含$r$分量的简单无穷级数。特别地，我们的结果推广和细化了由breyer - major, Giraitis-Surgailis和Arcones给出的关于高斯从属时间序列部分和的正态逼近的经典定理。
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