Comparison between criteria leading to the weak invariance principle

TL;DR

This paper compares different criteria for the central limit theorem and weak invariance principle, showing that in ergodic systems with positive entropy, these criteria can differ for certain functions.

Contribution

It provides a comparative analysis of multiple criteria leading to the weak invariance principle, highlighting their differences in ergodic systems with positive entropy.

Findings

Criteria can differ for the same function in ergodic systems.

Some functions satisfy one criterion but not another.

The comparison clarifies the relationships between different conditions.

Abstract

The aim of this paper is to compare various criteria leading to the central limit theorem and the weak invariance principle. These criteria are the martingale-coboundary decomposition developed by Gordin in Dokl. Akad. Nauk SSSR 188 (1969), the projective criterion introduced by Dedecker in Probab. Theory Related Fields 110 (1998), which was subsequently improved by Dedecker and Rio in Ann. Inst. H. Poincar\'{e} Probab. Statist. 36 (2000) and the condition introduced by Maxwell and Woodroofe in Ann. Probab. 28 (2000) later improved upon by Peligrad and Utev in Ann. Probab. 33 (2005). We prove that in every ergodic dynamical system with positive entropy, if we consider two of these criteria, we can find a function in $\mathbb{L}^2$ satisfying the first but not the second.