High-dimensional central limit theorems for homogeneous sums

TL;DR

This paper extends high-dimensional central limit theorems to homogeneous sums, providing quantitative bounds that depend on moments and influences, with applications to universality and fourth moment phenomena.

Contribution

It offers a new quantitative high-dimensional CLT for homogeneous sums, improving existing results with polynomial dependence on the logarithm of the dimension.

Findings

Established bounds on the Kolmogorov distance depending on moments and influences

Derived high-dimensional versions of fourth moment and universality theorems

Sharpened existing quantitative CLTs using the new bounds

Abstract

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.