Some remarks on Gordin-Lif\v{s}ic's condition for martingale approximations

TL;DR

This paper investigates the Gordin-Lifsic condition for martingale approximations in stationary sequences, providing new sufficient conditions, discussing their optimality, and applying results to semi-linear processes.

Contribution

It offers new sufficient and necessary conditions for L2 martingale approximations under the Gordin-Lifsic condition, extending the CLT framework for stationary sequences.

Findings

Various L2 approximation types are characterized.

Conditions for CLT variants are established.

Application to semi-linear processes demonstrates practical relevance.

Abstract

In this note, we study a condition introduced by Gordin and Lif{\v s}ic in 1981 to establish the Central Limit Theorem for additive functionals of stationary Markov chains with normal transition operator. In the more general setting of strictly stationary sequences satisfying the Gordin-Lif{\v s}ic condition, we give sufficient (and sometimes also necessary) conditions for partial sums to be approximated in L2 by a martingale with stationary increments. Various types of L2 approximations are described, leading to different versions of the central limit theorem (annealed, quenched, functional form...). The optimality of the conditions is discussed, and an application to the class of semi-linear processes is presented.