Exponential inequalities for unbounded functions of geometrically ergodic Markov chains: applications to quantitative error bounds for regenerative Metropolis algorithms

• Published in: Statistics (Berlin, DDR) Vol. 51; no. 1; pp. 222 - 234
• Main Author: Wintenberger, Olivier
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 02.01.2017; Taylor & Francis Ltd; Taylor & Francis: STM, Behavioural Science and Public Health Titles
• Subjects: Algorithms; Computer simulation; confidence interval; Confidence intervals; Ergodic processes; exponential inequalities; Inequalities; Markov analysis; Markov chains; Mathematics; Metropolis algorithm; Monte Carlo simulation; Regeneration; Statistics; Metropolis algorithm; Markov chains; exponential inequalities; Confidence inter-val
• ISSN: 0233-1888; 1026-7786; 1029-4910; 1029-4910
• DOI: 10.1080/02331888.2016.1268205

The aim of this note is to investigate the concentration properties of unbounded functions of geometrically ergodic Markov chains. We derive concentration properties of centred functions with respect to the square of Lyapunov's function in the drift condition satisfied by the Markov chain. We apply the new exponential inequalities to derive confidence intervals for Markov Chain Monte Carlo algorithms. Quantitative error bounds are provided for the regenerative Metropolis algorithm of [Brockwell and Kadane Identification of regeneration times in MCMC simulation, with application to adaptive schemes. J Comput Graphical Stat. 2005;14(2)].