A multivariate Berry–Esseen theorem for time-dependent expanding dynamical systems

The paper (v3) establishes the convex-set Berry–Esseen bound via a Bentkus–Fang style smoothing/induction and a precise leave-one-out Stein decomposition, with assumptions (UE:1–3) and blockwise covariance conditions (C1)–(C2), culminating in Theorem 2.2’s bound; the author explicitly notes a corrected decomposition in Section 4.3 relative to earlier drafts. The model’s alternative martingale–coboundary route is not justified at key steps: it assumes uniform conditional-expectation decay without proof, asserts a uniform-in-N covariance replacement ΣN ≈ ΓN without controlling cross-covariances, and treats the boundary term RN through an invalid direct triangle inequality for the convex-set metric. It also misattributes the +2(C0′)^{3/2} term to smoothing rather than to the blockwise trivial bound used in the paper. Hence, while the final rate matches the paper’s statement, the model’s proof outline has serious gaps.