October 2023 Vector-valued statistics of binomial processes: Berry–Esseen bounds in the convex distance
Mikołaj J. Kasprzak, Giovanni Peccati
Author Affiliations +
Ann. Appl. Probab. 33(5): 3449-3492 (October 2023). DOI: 10.1214/22-AAP1897


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.


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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


Received: 1 April 2022; Revised: 1 August 2022; Published: October 2023
First available in Project Euclid: 3 November 2023

Digital Object Identifier: 10.1214/22-AAP1897

Primary: 60B10 , 60F05
Secondary: 60D05 , 60E05 , 60E15

Keywords: Binomial process , Boolean model , convex distance , Stein’s method

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 5 • October 2023
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