An explicit Berry-Ess\'een bound for uniformly expanding maps on the interval

TL;DR

This paper provides an explicit Berry-Esséen bound with a concrete constant for uniformly expanding maps on the interval, improving upon previous bounds that lacked explicit constants.

Contribution

The authors introduce a new method using complex cone techniques to derive explicit Berry-Esséen bounds for expanding maps, applicable beyond just interval maps.

Findings

Established an explicit Berry-Esséen bound with a specified constant.

Demonstrated the method's applicability to a broader class of systems.

Improved the theoretical understanding of convergence rates in dynamical systems.

Abstract

For uniformly expanding maps on the interval, analogous versions of the Berry-Ess\'een theorem are known but only with an unexplicit upper bound in $O(1/\sqrt{n})$ without any constants being specified. In this paper, we use the recent complex cone technique to prove an explicit Berry-Ess\'een estimate with a reasonable constant for these maps. Our method is not limited to maps on the interval however and should apply to many situations.