A Multivariate CLT for Bounded Decomposable Random Vectors with the Best Known Rate

We prove a multivariate central limit theorem with explicit error bound in a non-smooth function distance for sums of bounded decomposable $$d$$ d -dimensional random vectors. The decomposition structure is similar to that of Barbour et al. (J Combin Theory Ser 47:125–145, 1989) and is more general than the local dependence structure considered in Chen and Shao (Ann Probab 32:1985–2028, 2004). The error bound is of the order $$d^{\frac{1}{4}} n^{-\frac{1}{2}}$$ d 1 4 n - 1 2 , where $$d$$ d is the dimension and $$n$$ n is the number of summands. The dependence on $$d$$ d , namely $$d^{\frac{1}{4}}$$ d 1 4 , is the best known dependence even for sums of independent and identically distributed random vectors, and the dependence on $$n$$ n , namely $$n^{-\frac{1}{2}}$$ n - 1 2 , is optimal. We apply our main result to a random graph example.