High-Dimensional Central Limit Theorems for Homogeneous Sums

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