Incorporating Cramer's condition into Stein's method for non-smooth test functions

Develop a Stein-method-based framework that directly incorporates Cramer's condition to handle non-smooth test functions in deriving Edgeworth-type results for sums of high-dimensional independent random vectors. The goal is to rigorously replace smoothness assumptions on the test function by Cramer's condition within Stein's method, thereby enabling expansions for indicators of rectangles and related sets.

Background

In the fixed-dimensional Fourier-analytic approach, Cramer's condition is routinely used to drop smoothness requirements on test functions when deriving Edgeworth expansions. However, the paper relies on Stein's method to obtain valid high-dimensional expansions for probabilities over rectangles and notes a fundamental methodological gap: Stein's method currently does not directly accommodate Cramer's condition in place of smoothness.

Bridging this gap would broaden the applicability of Stein's method to non-smooth functionals that are central in high-dimensional inference, such as maxima and rectangular events.