A Lyapunov-type Bound in Rd

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High-dimensional central limit theorems by Stein’s method (2020, background): “Bentkus (2005) proved that, with Z ∼ N(0, ), (1.9) sup h=1A:A∈C ∣∣Eh(W) − Eh(Z)∣∣ ≤ Cd1/4 n3/2 n∑ i=1 E [∣∣ −1/2Xi ∣∣3], where C is the collection of all (measurable) convex sets of Rd and | · | denotes the Euclidean norm when applied to a vector.”
High-dimensional econometrics and regularized GMM (2018, background): “For example, [28, 29] show that a result like (2.”
Nonparametric Tests for Treatment Effect Heterogeneity (2008, unknown): “Then, by Theorem 1.1, Bentkus (2005), where A K is the class of all measurable convex sets in K-dimensional Euclidean space, C is an absolute constant, and Φ is a multivariate standard Gaussian distribution.”
Large ball probabilities, Gaussian comparison and anti-concentration (2017, background): “It is known (see Bentkus (2005)[Theorem 1.2]) that if C satisﬁes conditions (i), (ii) and (1.9) then for some absolute constant C one has (1.10) Therefore, the inequalities (1.9), i.e. knowledge of a p , play the crucial role in the form of the bound (1.10).”