Randomized quasi-Monte Carlo, via certain scramblings of digital nets, produces unbiased estimates of $\int_{[0,1]^d}f(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x}$ with a variance that is $o(1/n)$ for any $f\in L^2[0,1]^d$. It also satisfies some non-asymptotic bounds where the variance is no larger than some $\Gamma<\infty$ times the ordinary Monte Carlo variance. For scrambled Sobol' points, this quantity $\Gamma$ grows exponentially in $d$. For scrambled Faure points, $\Gamma \leqslant \exp(1)\doteq 2.718$ in any dimension, but those points are awkward to use for large $d$. This paper shows that certain scramblings of Halton sequences have gains below an explicit bound that is $O(\log d)$ but not $O( (\log d)^{1-\epsilon})$ for any $\epsilon>0$ as $d\to\infty$. For $6\leqslant d\leqslant 10^6$, the upper bound on the gain coefficient is never larger than $3/2+\log(d/2)$.

翻译：随机拟蒙特卡洛方法通过对数字网络进行特定加扰，可对$\int_{[0,1]^d}f(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x}$产生无偏估计，且对于任意$f\in L^2[0,1]^d$，其方差为$o(1/n)$。该方法还满足某些非渐近界，方差不超过普通蒙特卡洛方差的某个有限倍数$\Gamma<\infty$。对于加扰的索博尔点，此量$\Gamma$随$d$呈指数增长；对于加扰的福尔点，$\Gamma \leqslant \exp(1)\doteq 2.718$在任何维度成立，但这类点在高维场景中难以应用。本文证明，哈尔顿序列的特定加扰具有低于显式界$O(\log d)$的增益，且当$d\to\infty$时，该界对任意$\epsilon>0$不会达到$O( (\log d)^{1-\epsilon})$的量级。对于$6\leqslant d\leqslant 10^6$的维度，增益系数的上界始终不高于$3/2+\log(d/2)$。