Before and beyond the mixing time: New approximations for additive functionals of stationary Gauss-Markov processes

TL;DR

This paper develops new limit theorems for additive functionals of stationary Gaussian Markov processes, especially around the critical mixing time scale, revealing phase transitions and new limit behaviors.

Contribution

It introduces quantitative and functional limit theorems for Gaussian Markov processes across various mixing time scales, highlighting phase transitions at the critical point.

Findings

Identifies a phase transition at the mixing time scale $t_{mix} \\asymp n$.

Derives new limit processes before, at, and after the mixing time.

Provides a comprehensive understanding of additive functionals in different mixing regimes.

Abstract

Whereas classical invariance principles for ergodic Markov chains address the situation in which the time horizon of observations is much larger than the mixing time, the quality of approximation is questionable when this is not the case anymore -- even when starting the Markov chain in the invariant law. In this article, we prove quantitative and functional limit theorems for additive functionals along triangular arrays of stationary Gaussian Markov processes when the mixing time $t_{\text{mix}}$ scales sub-, super- and proportionately to the number of observations $n$. Our major finding is a phase-transition at $t_{\text{mix}}\asymp n$, together with the identification and interrelation properties of the emerging new limit processes at and before the mixing time.