Gain Coefficients for Scrambled Halton Points

SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1021-1038, June 2024.
Abstract. Randomized quasi-Monte Carlo, via certain scramblings of digital nets, produces unbiased estimates of [math] with a variance that is [math] for any [math]. It also satisfies some nonasymptotic bounds where the variance is no larger than some [math] times the ordinary Monte Carlo variance. For scrambled Sobol’ points, this quantity [math] grows exponentially in [math]. For scrambled Faure points, [math] in any dimension, but those points are awkward to use for large [math]. This paper shows that certain scramblings of Halton sequences have gains below an explicit bound that is [math] but not [math] for any [math] as [math]. For [math], the upper bound on the gain coefficient is never larger than [math].