Skewness of a randomized quasi-Monte Carlo estimate

Some recent work on confidence intervals for randomized quasi-Monte Carlo (RQMC) sampling found a surprising result: ordinary Student's t 95% confidence intervals based on a modest number of replicates were seen to be very effective and even more reliable than some bootstrap t intervals that were expected to be best. One potential explanation is that those RQMC estimates have small skewness. In this paper we give conditions under which the skewness is $O\left(n^{ϵ}\right)$ for any $ϵ>0$, so ‘almost $O\left(1\right)$’. Under a random generator matrix model, we can improve this rate to $O\left(n^{-1/2+ϵ}\right)$ with very high probability. We also improve some probabilistic bounds on the distribution of the quality parameter t for a digital net in a prime base under random sampling of generator matrices.