Berry-Esseen bound for the Brownian motions on hyperbolic spaces

TL;DR

This paper establishes the rate at which the radial part of Brownian motion on hyperbolic spaces converges to a Gaussian distribution, providing sharp bounds in certain dimensions using advanced formulas.

Contribution

It derives the uniform convergence rate for Gaussian fluctuations of Brownian motion on hyperbolic spaces, with sharp results in specific dimensions, utilizing Millson and integration by parts formulas.

Findings

Established the convergence rate for the radial part of Brownian motion.

Proved the sharpness of the rate in two and odd dimensions.

Applied advanced formulas to analyze Gaussian fluctuations.

Abstract

We obtain the uniform convergence rate for the Gaussian fluctuation of the radial part of the Brownian motion on a hyperbolic space. We also show that this result is sharp if the dimension of the hyperbolic space is two or general odd. Our approach is based on the repetitive use of the Millson formula and the integration by parts formula.