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 $\delta >0$.

24 pages

References in corpus (7)

• Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction
• Optimal Hoeffding bounds for discrete reversible Markov chains
• Consistency of Markov chain quasi-Monte Carlo on continuous state spaces
• Variance bounding Markov chains
• Construction of weakly CUD sequences for MCMC sampling
• Discrepancy estimates for variance bounding Markov chain quasi-Monte Carlo
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Cited by in corpus (5)

• Discrepancy estimates for variance bounding Markov chain quasi-Monte Carlo
• Computation of expectations by Markov chain Monte Carlo methods
• A table of short-period Tausworthe generators for Markov chain quasi-Monte Carlo
• Improved bounds for the bracketing number of orthants or revisiting an algorithm of Thiémard to compute bounds for the star discrepancy
• A search for short-period Tausworthe generators over ${\mathbb{F}}_b$ with application to Markov chain quasi-Monte Carlo