Abstract:This note corrects several statements and proof steps in J.~Dick, \emph{Annals of Statistics} \textbf{39} (2011), 1372--1398, DOI: https://doi.org/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 $O(N^{-\min(\alpha,d)-1/2+\varepsilon})$ for every $\varepsilon>0$ 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nor directly controls the finite differences used in Appendix~B; the claimed theorem for that variation is therefore withdrawn here.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.
From: Josef Dick [view email] [via VTEX proxy]
[v1]
Tue, 6 Jul 2010 09:30:11 UTC (36 KB)
[v2]
Wed, 7 Jul 2010 05:04:24 UTC (36 KB)
[v3]
Thu, 8 Jul 2010 00:14:31 UTC (36 KB)
[v4]
Tue, 20 Nov 2012 09:12:28 UTC (103 KB)
[v5]
Tue, 25 Aug 2026 08:34:01 UTC (17 KB)
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