Cubic-Root Gaussian Approximation under Unrestricted Covariance
Abstract
For Gaussian approximation over high-dimensional rectangles under unrestricted covariance, Chernozhukov et al. (2023b) conjectured that the n−1/4 rate, up to logarithmic factors, is near-optimal. We show that, under the coordinatewise subexponential condition with scale Bn and the marginal variance lower bound condition with constant b in Chernozhukov et al. (2023b), the approximation error in dimension d is bounded by Cbmin{ 1, (Bn2/n)1/3{log(2dn)}7/3 + Bn/√n{log(2dn)}5/2 }. In particular, for bounded Bn and polynomial dimension, the new bound is n−1/3 and therefore falsifies the polynomial-dimensional n−1/4 near-optimality conjecture. The proof uses a two-stage interpolation and a rank-free matrix-weighted Gaussian surface bound, which may be of independent interest. The initial proof attempt was generated by ChatGPT 5.6 Pro (OpenAI) and subsequently corrected and rewritten by the authors. The machine-checked Lean formalization of the proof can be found at the GitHub repository (https://github.com/WeihanZhang2001/cubic-root-gaussian-approximation-under-unrestricted-covariance).
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BibTeX
@misc{gao2026cubicroot,
title = {{Cubic-Root Gaussian Approximation under Unrestricted Covariance}},
author = {Zijun Gao and Weihan Zhang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30221},
eprint = {2608.30221},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30221},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cubic-Root Gaussian Approximation under Unrestricted Covariance
AU - Zijun Gao
AU - Weihan Zhang
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30221
UR - https://arxiv.org/abs/2608.30221
AB - For Gaussian approximation over high-dimensional rectangles under unrestricted covariance, Chernozhukov et al. (2023b) conjectured that the n^−1/4 rate, up to logarithmic factors, is near-optimal. We show that, under the coordinatewise subexponential condition with scale B_n and the marginal variance lower bound condition with constant b in Chernozhukov et al. (2023b), the approximation error in dimension d is bounded by C_bmin{ 1, (B_n^2/n)^1/3{log(2dn)}^7/3 + B_n/√n{log(2dn)}^5/2 }. In particular, for bounded B_n and polynomial dimension, the new bound is n^−1/3 and therefore falsifies the polynomial-dimensional n^−1/4 near-optimality conjecture. The proof uses a two-stage interpolation and a rank-free matrix-weighted Gaussian surface bound, which may be of independent interest. The initial proof attempt was generated by ChatGPT 5.6 Pro (OpenAI) and subsequently corrected and rewritten by the authors. The machine-checked Lean formalization of the proof can be found at the GitHub repository (https://github.com/WeihanZhang2001/cubic-root-gaussian-approximation-under-unrestricted-covariance).
ER -