Highdimensional central limit theorems for homogeneous sums
Abstract
This paper develops a quantitative version of de Jong's central limit theorem for homogeneous sums in a highdimensional setting. More precisely, under appropriate moment assumptions, we establish an upper bound for the Kolmogorov distance between a multidimensional 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 highdimensional versions of fourth moment theorems, universality results and PeccatiTudor type theorems for homogeneous sums. We also sharpen some existing (quantitative) central limit theorems by applications of our result.
 Publication:

arXiv eprints
 Pub Date:
 February 2019
 arXiv:
 arXiv:1902.03809
 Bibcode:
 2019arXiv190203809K
 Keywords:

 Mathematics  Probability;
 Mathematics  Statistics Theory;
 60F05;
 62E17;
 47D07
 EPrint:
 49 pages. Some errors have been corrected. Application to a statistical problem has been added