Exactness of martingale approximation and the central limit theorem

TL;DR

This paper examines the precise conditions under which certain central limit theorems for Markov chains hold, demonstrating the necessity of assumptions like normality and extending convergence results to different laws.

Contribution

It establishes the sharpness of key CLTs for Markov chains, showing the necessity of assumptions and improving convergence results with new examples.

Findings

Normality assumption in Derriennic and Lin's CLT is essential.

Sharpness of Kipnis-Varadhan and Maxwell-Woodroofe CLTs demonstrated.

Extended convergence to different laws in specific cases.

Abstract

The article is showing sharpness of central limit theorems of Kipnis and Varadhan, Derriennic and Lin, Maxwell and Woodroofe. In the case of the CLT of Derriennic and Lin (for Markov chains with a normal operator) it is shown that the assumption of normality cannot be relaxed. In the case of the CLT of Maxwell and Woodroofe, the example of Peligrad and Utev is improved in the sense of getting a convergence to different laws.