Improved Berry-Esseen bounds for multivariate nonlinear statistics in convex distance
Abstract
In this paper, we establish two nonasymptotic Berry--Esseen bounds over convex sets for the Gaussian approximation of multivariate nonlinear statistics. The statistics of interest can be written as a sum of independent centered random vectors plus a remainder that may depend on all observations. The first bound retains the classical factor d1/4 in the contribution of the independent sum, where d is the dimension, while controlling the remainder through its size and its sensitivity to replacing a single observation. The second bound expresses the contribution of the independent sum in terms of fourth moments and can allow the dimension to grow faster with the sample size. For sums of independent random vectors, it removes the logarithmic factor from an existing fourth moment bound without imposing additional moment assumptions. As applications, we apply these results to Polyak--Ruppert averaging for nonsmooth stochastic approximation, temporal difference learning with linear function approximation, and multivariate U-statistics. The resulting bounds provide explicit Gaussian approximation errors and sufficient conditions under which these errors converge to zero as the dimension grows with the sample size.
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BibTeX
@misc{cai2026improved,
title = {{Improved Berry-Esseen bounds for multivariate nonlinear statistics in convex distance}},
author = {Zhi-Jun Cai and Jing-Cai Yang and Zhuo-Song Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10022},
eprint = {2610.10022},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10022},
}
RIS
TY - UNPB
TI - Improved Berry-Esseen bounds for multivariate nonlinear statistics in convex distance
AU - Zhi-Jun Cai
AU - Jing-Cai Yang
AU - Zhuo-Song Zhang
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10022
UR - https://arxiv.org/abs/2610.10022
AB - In this paper, we establish two nonasymptotic Berry--Esseen bounds over convex sets for the Gaussian approximation of multivariate nonlinear statistics. The statistics of interest can be written as a sum of independent centered random vectors plus a remainder that may depend on all observations. The first bound retains the classical factor d^1/4 in the contribution of the independent sum, where d is the dimension, while controlling the remainder through its size and its sensitivity to replacing a single observation. The second bound expresses the contribution of the independent sum in terms of fourth moments and can allow the dimension to grow faster with the sample size. For sums of independent random vectors, it removes the logarithmic factor from an existing fourth moment bound without imposing additional moment assumptions. As applications, we apply these results to Polyak--Ruppert averaging for nonsmooth stochastic approximation, temporal difference learning with linear function approximation, and multivariate U-statistics. The resulting bounds provide explicit Gaussian approximation errors and sufficient conditions under which these errors converge to zero as the dimension grows with the sample size.
ER -