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 $delta>0$.
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机译：马尔可夫链蒙特卡罗（MCMC）仿真的模型是由真实随机数驱动的。我们考虑由确定性数字序列驱动的方差边界马尔可夫链。恒星偏差提供了这种马尔可夫链准蒙特卡罗方法的效率度量。我们定义了驱动程序序列的回退差异，并指出了与马尔可夫链准蒙特卡洛样本的恒星差异密切相关。我们证明存在确定性的驱动器序列，使得差异几乎随着蒙特卡洛比率$ n ^ {-1/2} $减小。对于MCMC模拟，还可以考虑将马尔可夫链准蒙特卡洛的老化期考虑在内，以减少初始状态的影响。特别是，我们的差异范围会导致对期望计算的误差的估计。为了说明我们的理论，我们提供了一个基于球形步行的Metropolis算法的示例。此外，在附加的假设下，我们证明了驱动程序序列的存在，以使得对应的确定性马尔可夫链样本的差异随着$ n ^ {-1+ delta} $的顺序而降低，每$ delta> 0 $。
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