We establish an invariance principle for polynomial functions of $n$ independent high-dimensional random vectors, and also show that the obtained rates are nearly optimal. Both the dimension of the vectors and the degree of the polynomial are permitted to grow with $n$. Specifically, we obtain a finite sample upper bound for the error of approximation by a polynomial of Gaussians, measured in Kolmogorov distance, and extend it to functions that are approximately polynomial in a mean squared error sense. We give a corresponding lower bound that shows the invariance principle holds up to polynomial degree $o(\log n)$. The proof is constructive and adapts an asymmetrisation argument due to V. V. Senatov. As applications, we obtain a higher-order delta method with possibly non-Gaussian limits, and generalise a number of known results on high-dimensional and infinite-order U-statistics, and on fluctuations of subgraph counts.

翻译：我们针对$n$个独立高维随机向量的多项式函数建立了一个不变性原理，并证明所得到的估计速率接近最优。向量维度及多项式次数均可随$n$增长而增长。具体而言，我们获得了基于Kolmogorov距离度量的高斯多项式逼近误差的有限样本上界，并将其推广至均方误差意义下近似多项式的函数。我们给出了相应的下界，表明不变性原理在多项式次数为$o(\log n)$时成立。该证明具有构造性，并借鉴了V. V. Senatov的非对称化论证。作为应用，我们得到了一个允许非高斯极限的高阶delta方法，并推广了关于高维与无穷阶U统计量以及子图计数波动性的若干已知结论。