This paper develops a quantitative version of de Jong's central limit theorem for homogeneous sums in a high-dimensional setting. More precisely, under appropriate moment assumptions, we establish an upper bound for the Kolmogorov distance between a multi-dimensional vector of homogeneous sums and a Gaussian vector so that the bound depends polynomially on the logarithm of the dimension and is governed by the fourth cumulants and the maximal influences of the components. As a corollary, we obtain high-dimensional versions of fourth moment theorems, universality results and Peccati-Tudor type theorems for homogeneous sums. We also sharpen some existing (quantitative) central limit theorems by applications of our result.

翻译：本文为高维环境中的同质总量开发了德钟中央限制理论的定量版本。 更准确地说, 在适当时刻的假设下, 我们为同质总量和高西亚矢量的多维矢量之间的 Kolmogorov 距离设定了一个上限, 以便该约束线在多维矢量上依赖维度的对数, 并受第四层蓄积量和各个组成部分的最大影响所制约。 作为必然结果, 我们获得了第四刻论、 普遍性结果和Pecati- Tudor 类型等量的高维版本。 我们还通过应用我们的结果来强化一些现有的( Q) 中心限制理论。