Discrepancy bounds for uniformly ergodic Markov chain quasi-Monte Carlo
Provides theoretical guarantees for using quasi-Monte Carlo methods in Markov chain sampling, benefiting researchers in computational statistics and Monte Carlo methods.
The paper proves upper bounds on the discrepancy between the empirical distribution of samples from a uniformly ergodic Markov chain and the target distribution when using deterministic driver sequences, showing that the discrepancy can converge at nearly the standard Monte Carlo rate of n^{-1/2}.
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 $n^{-1/2}$.