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。