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This paper provides a finite sample bound for the error term in the Edgeworthexpansion for a sum of independent, potentially discrete, nonlattice randomvectors, using a uniform-in-$P$ version of the weaker Cram\'{e}r condition inAngst and Poly (2017). This finite sample bound can be used to derive anEdgeworth expansion that is uniform over the distributions of the randomvectors. Using this result, we derive a uniform-in-$P$ higher order expansionof resampling-based distributions.

A Uniform-in-$P$ Edgeworth Expansion under Weak Cram\'{e}r Conditions