On the functional central limit theorem via martingale approximation

TL;DR

This paper establishes necessary and sufficient conditions for martingale approximation of stationary processes, facilitating the transfer of functional central limit theorems and unifying various classes of stochastic processes.

Contribution

It provides a simple, adaptable condition for martingale approximation applicable to many stochastic processes, enhancing understanding of their asymptotic behavior.

Findings

Condition valid for Maxwell–Woodroofe class

Applicable to strongly mixing processes

Unifies various stochastic process examples

Abstract

In this paper, we develop necessary and sufficient conditions for the validity of a martingale approximation for the partial sums of a stationary process in terms of the maximum of consecutive errors. Such an approximation is useful for transferring the conditional functional central limit theorem from the martingale to the original process. The condition found is simple and well adapted to a variety of examples, leading to a better understanding of the structure of several stochastic processes and their asymptotic behaviors. The approximation brings together many disparate examples in probability theory. It is valid for classes of variables defined by familiar projection conditions such as the Maxwell--Woodroofe condition, various classes of mixing processes, including the large class of strongly mixing processes, and for additive functionals of Markov chains with normal or symmetric Markov operators.