Complete optimality characterization in the independent-vector specialization under minimal moments

Let $n,d\in\mathbb N$. For each $i\in\{1,\dots,n\}$, let $X_i=(X_{i1},\dots,X_{id})^\top\in\mathbb R^d$ be independent random vectors with $\mathbb E[X_i]=0$. Define the normalized sum

its covariance matrix

and let $Z_n\sim N(0,\Sigma_n)$ be the centered Gaussian vector with the same covariance. Let $\mathcal H_d$ be the class of axis-aligned hyperrectangles in $\mathbb R^d$, e.g.

For a given law $P$ of $(X_1,\dots,X_n)$, define the rectangle Kolmogorov distance

This setup follows Bong et al. (2025).

Assume finite third moments and coordinatewise nondegeneracy: for all $i,j$, $\mathbb E|X_{ij}|^3<\infty$, and there is a constant $\underline\sigma>0$ such that $\min_{1\le j\le d}(\Sigma_n)_{jj}\ge \underline\sigma^2$. For $B<\infty$, define

and minimax risk