A Uniform-in-$P$ Edgeworth Expansion under Weak Cramér Conditions

This paper provides a finite sample bound for the error term in the Edgeworth expansion for a sum of independent, potentially discrete, nonlattice random vectors, using a uniform-in-$P$ version of the weaker Cram\'{e}r condition in Angst and Poly(2017). This finite sample bound is used to derive a bound for the error term in the Edgeworth expansion that is uniform over the joint distributions $P$ of the random vectors, and eventually to derive a higher order expansion of resampling-based distributions in a unifying way. As an application, we derive a uniform-in-$P$ Edgeworth expansion of bootstrap distributions and that of randomized subsampling distributions, when the joint distribution of the original sample is absolutely continuous with respect to Lebesgue measure.