The nonzero gain coefficients of Sobol's sequences are always powers of two

Abstract

When a plain Monte Carlo estimate on $n$ samples has variance $\sigma^2/n$, then scrambled digital nets attain a variance that is $o(1/n)$ as $n\to\infty$. For finite $n$ and an adversarially selected integrand, the variance of a scrambled $(t,m,s)$-net can be at most $\Gamma\sigma^2/n$ for a maximal gain coefficient $\Gamma<\infty$. The most widely used digital nets and sequences are those of Sobol'. It was previously known that $\Gamma\leqslant 2^t3^s$ for Sobol' points as well as Niederreiter-Xing points. In this paper we study nets in base $2$. We show that $\Gamma \leqslant2^{t+s-1}$ for nets. This bound is a simple, but apparently unnoticed, consequence of a microstructure analysis in Niederreiter and Pirsic (2001). We obtain a sharper bound that is smaller than this for some digital nets. We also show that all nonzero gain coefficients must be powers of two. A consequence of this latter fact is a simplified algorithm for computing gain coefficients of nets in base $2$.