Super-polynomial accuracy of one dimensional randomized nets using the median of means

Abstract

Letffbe analytic on[0,1][0,1]with|f(k)(1/2)|⩽Aαkk!|f^{(k)}(1/2)|\leqslant A\alpha ^kk!for some constantsAAandα>2\alpha >2and allk⩾1k\geqslant 1. We show that the median estimate ofμ=∫01f(x)dx\mu =\int _0^1f(x)\,\mathrm {d} xunder random linear scrambling withn=2mn=2^mpoints converges at the rateO(n−clog⁡(n))O(n^{-c\log (n)})for anyc>3log⁡(2)/π2≈0.21c> 3\log (2)/\pi ^2\approx 0.21. We also get a super-polynomial convergence rate for the sample median of2k−12k-1random linearly scrambled estimates, whenk/mk/mis bounded away from zero. Whenffhas app’th derivative that satisfies aλ\lambda-Hölder condition then the median of means has errorO(n−(p+λ)+ϵ)O( n^{-(p+\lambda )+\epsilon })for anyϵ>0\epsilon >0, ifk→∞k\to \inftyasm→∞m\to \infty. The proof techniques use methods from analytic combinatorics that have not previously been applied to quasi-Monte Carlo methods, most notably an asymptotic expression from Hardy and Ramanujan on the number of partitions of a natural number.