Some remarks on Gordin-Lifšic's condition for martingale approximations

2025/10/23 by Jérôme Dedecker, Dedecker, Jèrôme, Florence Merlevède +1
Business, Management and Accounting · Economics, Econometrics and Finance · Mathematics · #Advanced Queuing Theory Analysis #Stochastic processes and financial applications #Nonlinear Differential Equations Analysis

paper · pdf · doi:10.48550/arxiv.2510.20501

Abstract

In this note, we study a condition introduced by Gordin and Lif\v sic 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 sic 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.

Citations

• On the quenched CLT for stationary Markov chains
• A quenched weak invariance principle
• A quenched invariance principle for stationary processes

Related

• Quenched Invariance Principles via Martingale Approximation
• Martingale approximations for sums of stationary processes
• Exactness of martingale approximation and the central limit theorem
• Martingale approximation and optimality of some conditions for the central limit theorem
• Central limit theorem started at a point for additive functionals of reversible Markov chains