A quenched invariance principle for stationary processes

TL;DR

This paper establishes a quenched invariance principle for stationary processes under the Hannan condition, providing a new perspective by representing these processes as functionals of Markov chains.

Contribution

It introduces a conditionally centered quenched invariance principle and a novel construction linking stationary processes to Markov chains.

Findings

Proves a quenched weak invariance principle under the Hannan condition

Provides a new construction of stationary processes as Markov chain functionals

Enhances understanding of the structure of stationary processes

Abstract

In this note, we prove a conditionally centered version of the quenched weak invariance principle under the Hannan condition, for stationary processes. In the course, we obtain a (new) construction of the fact that any stationary process may be seen as a functional of a Markov chain.