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arXiv · 1902.03809

High-dimensional central limit theorems for homogeneous sums

Yuta Koike
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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.

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Yuta Koike. 2019-02-11. High-dimensional central limit theorems for homogeneous sums. https://arxiv.org/abs/1902.03809

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