## The Annals of Applied Probability

### Discrepancy bounds for uniformly ergodic Markov chain quasi-Monte Carlo

#### Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1476884315

Digital Object Identifier
doi:10.1214/16-AAP1173

Mathematical Reviews number (MathSciNet)
MR3563205

Zentralblatt MATH identifier
1351.60100

#### Citation

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. https://projecteuclid.org/euclid.aoap/1476884315

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