High-Dimensional Central Limit Theorems for Homogeneous Sums

Yuta Koike ()
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Yuta Koike: University of Tokyo and CREST JST

Journal of Theoretical Probability, 2023, vol. 36, issue 1, 1-45

Abstract: 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.

Keywords: de Jong’s theorem; Fourth-moment theorem; High dimensions; Peccati–Tudor-type theorem; Quantitative CLT; Randomized Lindeberg method; Stein kernel; Universality; 60F05; 62E17; 47D07 (search for similar items in EconPapers)
Date: 2023
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-022-01156-2

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