Rates of convergence for multivariate normal approximation with applications to dense graphs and doubly indexed permutation statistics

TL;DR

This paper introduces a new theorem for multivariate normal approximation using Stein couplings, with applications to dense graphs and permutation statistics, advancing the understanding of convergence rates in high-dimensional probability.

Contribution

It presents a novel general theorem for multivariate normal approximation on convex sets based on Stein couplings, applicable to complex combinatorial structures.

Findings

Established a new multivariate normal approximation theorem.

Applied the theorem to dense random graphs for homogeneity testing.

Proved asymptotic normality for specific doubly indexed permutation statistics.

Abstract

We provide a new general theorem for multivariate normal approximation on convex sets. The theorem is formulated in terms of a multivariate extension of Stein couplings. We apply the results to a homogeneity test in dense random graphs and to prove multivariate asymptotic normality for certain doubly indexed permutation statistics.