Phase transition of Schott's statistic for high-dimensional heavy-tailed data

[2606.12943] Phase transition of Schott's statistic for high-dimensional heavy-tailed data Skip to main content Search Submit Donate Log in Mathematics > Statistics Theory arXiv:2606.12943 (math) [Submitted on 11 Jun 2026 ( v1 ), last revised 27 Aug 2026 (this version, v2)] Title: Phase transition of Schott's statistic for high-dimensional heavy-tailed data Authors: Hantao Chen , Guangming Pan , Cheng Wang View PDF HTML (experimental) Abstract: Consider Schott's statistic (Schott, 2005) defined as the squared Frobenius norm of the sample correlation matrix for data from $\alpha$-regularly varying populations. We investigate its asymptotic distribution in a general framework characterized by data dimension p, sample size n, and regularly varying coefficients $\alpha$. In particular, we identify a phase transition phenomenon in the asymptotic behavior. For light-tailed populations ($\alpha > 3$), we revisit the $\alpha$-free asymptotic distribution but relax the constraint on the ratio of $p/n$. For heavy-tailed populations ($\alpha < 3$), we derive a new asymptotic normal distribution whose variance explicitly depends on $\alpha$. We also propose a consistent estimator for the asymptotic variance such that the standardized Schott's test statistic remains applicable for unknown location parameters and all $\alpha > 0$. Comments: 38 pages Subjects: Statistics Theory (math.ST) MSC classes: 62H10, 60F05 Cite as: arXiv:2606.12943 [math.ST] (or arXiv:2606.12943v2 [math.ST] for this version) https://doi.org/10.48550/arXiv.2606.12943 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Cheng Wang [ view email ] [v1] Thu, 11 Jun 2026 06:10:50 UTC (69 KB) [v2] Thu, 27 Aug 2026 05:58:16 UTC (25 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.ST < prev | next > new | recent | 2026-06 Change to browse by: math stat 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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