Abstract
We study the discrepancy between the distribution of a vector-valued functional of i.i.d. random elements and that of a Gaussian vector. Our main contribution is an explicit bound on the convex distance between the two distributions, holding in every dimension. Such a finding constitutes a substantial extension of the one-dimensional bounds deduced in Chatterjee (Ann. Probab. 36 (2008) 1584–1610) and Lachièze-Rey and Peccati (Ann. Appl. Probab. 27 (2017) 1992–2031), as well as of the multidimensional bounds for smooth test functions and indicators of rectangles derived, respectively, in Dung (Acta Math. Hungar. 158 (2019) 173–201), and Fang and Koike (Ann. Appl. Probab. 31 (2021) 1660–1686). Our techniques involve the use of Stein’s method, combined with a suitable adaptation of the recursive approach inaugurated by Schulte and Yukich (Electron. J. Probab. 24 (2019) 1–42): this yields rates of converge that have a presumably optimal dependence on the sample size. We develop several applications of a geometric nature, among which is a new collection of multidimensional quantitative limit theorems for the intrinsic volumes associated with coverage processes in Euclidean spaces.
Funding Statement
The authors were supported by the FNR grant FoRGES (R-AGR- 3376-10) at Luxembourg University.
This work is also part of project Stein-ML that has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska–Curie grant agreement No 101024264.
Citation
Mikołaj J. Kasprzak. Giovanni Peccati. "Vector-valued statistics of binomial processes: Berry–Esseen bounds in the convex distance." Ann. Appl. Probab. 33 (5) 3449 - 3492, October 2023. https://doi.org/10.1214/22-AAP1897
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