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

[1007.0842] Erratum to "Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands" Skip to main content Search Submit Donate Log in Mathematics > Numerical Analysis arXiv:1007.0842 (math) [Submitted on 6 Jul 2010 ( v1 ), last revised 25 Aug 2026 (this version, v5)] Title: Erratum to "Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands" Authors: Josef Dick View PDF HTML (experimental) 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nor directly controls the finite differences used in Appendix~B; the claimed theorem for that variation is therefore withdrawn here.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. Subjects: Numerical Analysis (math.NA) ; Statistics Theory (math.ST); Computation (stat.CO) Cite as: arXiv:1007.0842 [math.NA] (or arXiv:1007.0842v5 [math.NA] for this version) https://doi.org/10.48550/arXiv.1007.0842 Focus to learn more arXiv-issued DOI via DataCite Submission history 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) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.NA < prev | next > new | recent | 2010-07 Change to browse by: cs cs.NA math math.ST stat stat.CO stat.TH References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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