## Electronic Journal of Probability

### Discrepancy estimates for variance bounding Markov chain quasi-Monte Carlo

#### Abstract

Markov chain Monte Carlo (MCMC) simulations are modeled as driven by true random numbers. We consider variance bounding Markov chains driven by a deterministic sequence of numbers. The star-discrepancy provides a measure of efficiency of such Markov chain quasi-Monte Carlo methods. We define a pull-back discrepancy of the driver sequence and state a close relation to the star-discrepancy of the Markov chain-quasi Monte Carlo samples. We prove that there exists a deterministic driver sequence such that the discrepancies decrease almost with the Monte Carlo rate $n^{-1/2}$. As for MCMC simulations,  a burn-in period can also be taken into account for Markov chain quasi-Monte Carlo to reduce the influence of the initial state. In particular, our discrepancy bound leads to an estimate of the error for the computation of expectations. To illustrate our theory we provide an example for the Metropolis algorithm based on a ball walk. Furthermore, under additional assumptions we prove the existence of a driver sequence such that the discrepancy of the corresponding deterministic Markov chain sample decreases with order $n^{-1+\delta}$ for every $\delta>0$.

#### Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 105, 24 pp.

Dates
Accepted: 5 November 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065747

Digital Object Identifier
doi:10.1214/EJP.v19-3132

Mathematical Reviews number (MathSciNet)
MR3275857

Zentralblatt MATH identifier
1309.65004

Rights

#### Citation

Dick, Josef; Rudolf, Daniel. Discrepancy estimates for variance bounding Markov chain quasi-Monte Carlo. Electron. J. Probab. 19 (2014), paper no. 105, 24 pp. doi:10.1214/EJP.v19-3132. https://projecteuclid.org/euclid.ejp/1465065747

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