Optimal Bounds in Normal Approximation for Many Interacting Worlds

TL;DR

This paper employs Stein's method to derive optimal bounds in normal approximation for the empirical distribution of a many-interacting-worlds harmonic oscillator, confirming a conjecture and advancing theoretical understanding.

Contribution

It provides the first optimal bounds in Kolmogorov and Wasserstein distances for this model, resolving a conjecture and enhancing the theoretical framework.

Findings

Optimal bounds in Wasserstein and Kolmogorov distances

Resolution of McKeague and Levin's conjecture

Enhanced understanding of many-interacting-worlds harmonic oscillator

Abstract

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. (2014)]. Our bounds on the Wasserstein distance solve a conjecture of McKeague and Levin [Ann. Appl. Probab. (2016)].