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Discrepancy estimates for variance bounding Markov chain quasi-Monte Carlo

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1.	Title	Title of document	Discrepancy estimates for variance bounding Markov chain quasi-Monte Carlo
2.	Creator	Author's name, affiliation, country	Josef Dick; The Univesity of New South Wales; Australia
2.	Creator	Author's name, affiliation, country	Daniel Rudolf; Friedrich Schiller University of Jena; Germany
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Markov chain Monte Carlo ; Markov chain quasi-Monte Carlo ; variance bounding ; discrepancy theory ; spectral gap ; probabilistic method
3.	Subject	Subject classification	60J22; 65C40; 62F15; 65C05; 60J05
4.	Description	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 $\delta>0$.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	Australian Research Council, DFG
7.	Date	(YYYY-MM-DD)	2014-11-05
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/3132
10.	Identifier	Digital Object Identifier	10.1214/EJP.v19-3132
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 19
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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