The Annals of Applied Probability

Discrepancy bounds for uniformly ergodic Markov chain quasi-Monte Carlo

Josef Dick, Daniel Rudolf, and Houying Zhu

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Markov chains can be used to generate samples whose distribution approximates a given target distribution. The quality of the samples of such Markov chains can be measured by the discrepancy between the empirical distribution of the samples and the target distribution. We prove upper bounds on this discrepancy under the assumption that the Markov chain is uniformly ergodic and the driver sequence is deterministic rather than independent $U(0,1)$ random variables. In particular, we show the existence of driver sequences for which the discrepancy of the Markov chain from the target distribution with respect to certain test sets converges with (almost) the usual Monte Carlo rate of $n^{-1/2}$.

Article information

Ann. Appl. Probab., Volume 26, Number 5 (2016), 3178-3205.

Received: March 2013
Revised: February 2015
First available in Project Euclid: 19 October 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J22: Computational methods in Markov chains [See also 65C40] 65C40: Computational Markov chains 62F15: Bayesian inference
Secondary: 65C05: Monte Carlo methods 60J05: Discrete-time Markov processes on general state spaces

Markov chain Monte Carlo uniformly ergodic Markov chain discrepancy theory probabilistic method


Dick, Josef; Rudolf, Daniel; Zhu, Houying. Discrepancy bounds for uniformly ergodic Markov chain quasi-Monte Carlo. Ann. Appl. Probab. 26 (2016), no. 5, 3178--3205. doi:10.1214/16-AAP1173.

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