Strong Invariance Principles for Ergodic Markov Processes

TL;DR

This paper establishes strong invariance principles for ergodic Markov processes, enabling better analysis of estimators and simulation methods in dependent stochastic systems.

Contribution

It extends strong invariance principles to continuous-time ergodic Markov processes, providing a unified framework for analyzing estimators and simulation techniques.

Findings

Unified analysis of asymptotic variance estimators

Application to batch means method in Monte Carlo simulations

Fluctuation results for additive functionals of ergodic diffusions

Abstract

Strong invariance principles describe the error term of a Brownian approximation of the partial sums of a stochastic process. While these strong approximation results have many applications, the results for continuous-time settings have been limited. In this paper, we obtain strong invariance principles for a broad class of ergodic Markov processes. Strong invariance principles provide a unified framework for analysing commonly used estimators of the asymptotic variance in settings with a dependence structure. We demonstrate how this can be used to analyse the batch means method for simulation output of Piecewise Deterministic Monte Carlo samplers. We also derive a fluctuation result for additive functionals of ergodic diffusions using our strong approximation results.