Edgeworth Series for Lattice Distributions

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.