Integrability conditions on coboundary and transfer function for limit theorems

TL;DR

This paper establishes integrability conditions on coboundary functions that ensure various limit theorems, such as the weak invariance principle and law of iterated logarithm, hold for certain measure-preserving dynamical systems.

Contribution

It provides new criteria based on tail functions for coboundary and transfer functions to guarantee limit theorems in ergodic theory.

Findings

Conditions on tail functions ensure limit theorems for $f=m+g-g\circ T$

Results apply to systems with martingale difference sequences

Unified approach to multiple limit theorems in ergodic theory

Abstract

For a measure preserving automorphism $T$ of a probability space, we provide conditions on the tail function of $g\colon\Omega\to\mathbb R$ and $g-g\circ T$ which guarantee limit theorems among the weak invariance principle, Marcinkievicz-Zygmund strong law of large numbers and the law of iterated logarithm to hold for $f:=m+g-g\circ T$, where $(m\circ T^i)\_{i\geqslant 0}$ is a martingale differences sequence.