Quenched Invariance Principles via Martingale Approximation

TL;DR

This paper reviews the quenched invariance principles and almost sure central limit theorems for stationary ergodic processes, emphasizing martingale approximation techniques and their applications to various stochastic models.

Contribution

It provides a comprehensive survey of quenched invariance principles using martingale approximation for stationary ergodic processes and explores diverse applications.

Findings

Martingale approximation is key to proving quenched invariance principles.

The results apply to mixing sequences, shift processes, and reversible Markov chains.

The paper highlights the broad applicability of these theorems in stochastic processes.

Abstract

In this paper we survey the almost sure central limit theorem and its functional form (quenched) for stationary and ergodic processes. For additive functionals of a stationary and ergodic Markov chain these theorems are known under the terminology of central limit theorem and its functional form, started at a point. All these results have in common that they are obtained via a martingale approximation in the almost sure sense. We point out several applications of these results to classes of mixing sequences, shift processes, reversible Markov chains, Metropolis Hastings algorithms.