A uniform-in-P Edgeworth expansion under weak Cramér conditions

Abstract

This paper provides a finite sample bound for the error term in the Edgeworth expansion for a sum of independent, potentially discrete, non-lattice random vectors, using a uniform-in-P version of the weaker Cramér condition in [Angst J, Poly G. A Weak Cramèr Condition and Application to Edgeworth Expansions. Electronic J Prob. 2017;22:1–24]. This finite sample bound can be used to derive an Edgeworth expansion that is uniform over the distributions of the random vectors. Using this result, we derive a uniform-in-P higher-order expansion of resampling-based distributions.