Refinements in Asymptotic Expansions for Sums of Weakly Dependent Random Vectors

Abstract

Let $S_n$ denote the $n$th normalized partial sum of a sequence of mean zero, weakly dependent random vectors. This paper gives asymptotic expansions for $Ef(S_n)$ under weaker moment conditions than those of Gotze and Hipp (1983). It is also shown that an expansion for $Ef(S_n)$ with an error term $o(n^{-(s - 2)/2})$ is valid without any Cramer-type condition, if $f$ has partial derivatives of order $(s - 1)$ only. This settles a conjecture of Gotze and Hipp in their 1983 paper.