April 2023 Optimal bounds in normal approximation for many interacting worlds
Louis H. Y. Chen, Lê Vǎn Thành
Author Affiliations +
Ann. Appl. Probab. 33(2): 825-842 (April 2023). DOI: 10.1214/21-AAP1747


In this paper, we use Stein’s method to obtain optimal bounds, both in Kolmogorov and in Wasserstein distance, in the normal approximation for the empirical distribution of the ground state of a many-interacting-worlds harmonic oscillator proposed by Hall, Deckert and Wiseman (Phys. Rev. X 4 (2014) 041013). Our bounds on the Wasserstein distance solve a conjecture of McKeague and Levin (Ann. Appl. Probab. 26 (2016) 2540–2555).

Funding Statement

This research was partially supported by Grant R-146-000-230-114 from the National University of Singapore.


A substantial part of this paper was written when the second author was at the Vietnam Institute for Advanced Study in Mathematics (VIASM) and the Department of Mathematics, National University of Singapore (NUS). He would like to thank VIASM and the Department of Mathematics at NUS for their hospitality. We would also like to thank Adrian Röllin for helping us draw Figure 1 and the referees for their valuable comments.


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Louis H. Y. Chen. Lê Vǎn Thành. "Optimal bounds in normal approximation for many interacting worlds." Ann. Appl. Probab. 33 (2) 825 - 842, April 2023. https://doi.org/10.1214/21-AAP1747


Received: 1 June 2020; Revised: 1 April 2021; Published: April 2023
First available in Project Euclid: 21 March 2023

zbMATH: 07692277
MathSciNet: MR4564414
Digital Object Identifier: 10.1214/21-AAP1747

Primary: 60F05 , 81Q65

Keywords: harmonic oscillator , many interacting worlds , Normal approximation , Stein’s method

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 2 • April 2023
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