Open Access
2024 Strong invariance principles for ergodic Markov processes
Ardjen Pengel, Joris Bierkens
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Electron. J. Statist. 18(1): 191-246 (2024). DOI: 10.1214/23-EJS2199


Strong invariance principles describe the error term of a Brownian approximation to the partial sums of a stochastic process. While these strong approximation results have many applications, 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.


The authors thank the associate editor and the referees for their constructive comments and suggestions which helped to improve the manuscript. This work is part of the research programme ‘Zigzagging through computational barriers’ with project number 016.Vidi.189.043, which is financed by the Dutch Research Council (NWO).


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Ardjen Pengel. Joris Bierkens. "Strong invariance principles for ergodic Markov processes." Electron. J. Statist. 18 (1) 191 - 246, 2024.


Received: 1 June 2022; Published: 2024
First available in Project Euclid: 30 January 2024

Digital Object Identifier: 10.1214/23-EJS2199

Primary: 65C05
Secondary: 60J25

Keywords: asymptotic variance estimation , Piecewise deterministic Markov processes , Strong invariance principle

Vol.18 • No. 1 • 2024
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