A multivariate CLT for bounded decomposable random vectors with the best known rate

TL;DR

This paper establishes a multivariate central limit theorem with explicit error bounds for sums of bounded decomposable random vectors, improving the dependence on dimension and matching optimal rates.

Contribution

It introduces a new CLT for decomposable vectors with the best known dimension dependence and optimal sample size rate, extending previous local dependence results.

Findings

Error bound of order d^{1/4} n^{-1/2} for the CLT

Dependence on dimension is the best known

Dependence on sample size is optimal

Abstract

We prove a multivariate central limit theorem with explicit error bound on a non-smooth function distance for sums of bounded decomposable $d$-dimensional random vectors. The decomposition structure is similar to that of Barbour, Karo\'nski and Ruci\'nski (1989) and is more general than the local dependence structure considered in Chen and Shao (2004). The error bound is of the order $d^{\frac{1}{4}} n^{-\frac{1}{2}}$, where $d$ is the dimension and $n$ is the number of summands. The dependence on $d$, namely $d^{\frac{1}{4}}$, is the best known dependence even for sums of independent and identically distributed random vectors, and the dependence on $n$, namely $n^{-\frac{1}{2}}$, is optimal. We apply our main result to a random graph example.