Expansions in the local and the central limit theorems for dynamical systems

TL;DR

This paper investigates advanced probabilistic limit theorems for chaotic dynamical systems, providing higher-order expansions and broad applicability to various complex systems with unbounded observables.

Contribution

It introduces general results for Edgeworth and local limit theorem expansions in dynamical systems, extending classical probabilistic results to more complex, chaotic models.

Findings

Established higher-order expansion results under technical assumptions

Verified assumptions for various complex systems including Sinai billiards and random matrix products

Extended limit theorems to unbounded observables with near-optimal integrability conditions

Abstract

We study higher order expansions both in the Berry-Ess\'een estimate (Edgeworth expansions) and in the local limit theorems for Birkhoff sums of chaotic probability preserving dynamical systems. We establish general results under technical assumptions, discuss the verification of these assumptions and illustrate our results by different examples (subshifts of finite type, Young towers, Sinai billiards, random matrix products), including situations of unbounded observables with integrability order arbitrarily close to the optimal moment condition required in the i.i.d. setting.