Edgeworth Series for Lattice Distributions

Abstract

This paper investigates the use of Edgeworth expansions for approximating the distribution function of the normalized sum of $n$ independent and identically distributed lattice-valued random variables. We prove that the continuity-corrected Edgeworth series, using Sheppard-adjusted cumulants, is accurate to the same order in $n$ as the usual Edgeworth approximation for continuous random variables. Finally, as a partial justification of the Sheppard adjustments, it is shown that if a continuous random variable $Y$ is rounded into a discrete part $D$ and a truncation error $U$, such that $Y = D + U$, then under suitable limiting conditions the truncation error is approximately uniformly distributed and independent of $Y$, but not independent of $D$.