Erratum to "Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands"

Abstract

This note corrects several statements and proof steps in J.~Dick, \emph{Annals of Statistics} \textbf{39} (2011), 1372--1398, DOI: 10.1214/11-AOS880, arXiv:1007.0842 v4. The principal smooth-function result remains valid: an order-$d$ nested-uniformly scrambled digital $(t,m,ds)$-net has root mean square integration error $O(N^{-\min (\alpha ,d)-1/2+\epsilon })$ for every $\epsilon >0$ for functions in the unanchored Sobolev space with square-integrable mixed partial derivatives up to order $\alpha$ in each variable. The finite-difference variation introduced in the paper neither coincides with the displayed Sobolev norm nor directly controls the finite differences used in Appendix~B; the claimed theorem for that variation is therefore withdrawn here. In addition, the variance bound in Theorem~10 is missing a square, and the proof for $d>\alpha$ does not prove the logarithmic power printed in the theorem. We give a direct Sobolev proof, and a corrected logarithmic factor.