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

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, Karoński and Ruciński (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.

12 pages

References in corpus (2)

• Normal approximation under local dependence
• Stein couplings for normal approximation