A law of large numbers concerning the distribution of critical points of random Fourier series.
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| Title: | A law of large numbers concerning the distribution of critical points of random Fourier series. |
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| Authors: | |
| Source: | Stochastic Processes & Their Applications. May2026, Vol. 195, pN.PAG-N.PAG. 1p. |
| Subjects: | Law of large numbers, Random functions (Mathematics), Stochastic processes, Torus, Gaussian distribution, Asymptotic distribution |
| Abstract: | On the flat torus T m = R m / Z m we consider the Gaussian random function F a R defined as a random Fourier series (1.1). The Fourier coefficients are mean zero independent normal variables whose variances depend on the frequencies via an even Schwartz function a on R and large rescaling parameter R. For any open subset U of the torus denote by Z R (U) the number of critical points of F a R in U. We prove that if U is contained in a geodesic ball, then the variance of Z R (U) is asymptotic to const × Rmvol [ U ] as R → ∞. We use this to prove that if m ≥ 2, then as N → ∞, the random measures N − m Z N (−) converge a.s. to an explicit multiple of the volume measure on the flat torus. [ABSTRACT FROM AUTHOR] |
| Copyright of Stochastic Processes & Their Applications is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 191581394 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: A law of large numbers concerning the distribution of critical points of random Fourier series. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22"Brandon"+Fu%2C+Qiangang%22">"Brandon" Fu, Qiangang</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> qfu3@nd.edu</i><br /><searchLink fieldCode="AR" term="%22Nicolaescu%2C+Liviu+I%2E%22">Nicolaescu, Liviu I.</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> lnicolae@nd.edu</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Stochastic+Processes+%26+Their+Applications%22">Stochastic Processes & Their Applications</searchLink>. May2026, Vol. 195, pN.PAG-N.PAG. 1p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Law+of+large+numbers%22">Law of large numbers</searchLink><br /><searchLink fieldCode="DE" term="%22Random+functions+%28Mathematics%29%22">Random functions (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Stochastic+processes%22">Stochastic processes</searchLink><br /><searchLink fieldCode="DE" term="%22Torus%22">Torus</searchLink><br /><searchLink fieldCode="DE" term="%22Gaussian+distribution%22">Gaussian distribution</searchLink><br /><searchLink fieldCode="DE" term="%22Asymptotic+distribution%22">Asymptotic distribution</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: On the flat torus T m = R m / Z m we consider the Gaussian random function F a R defined as a random Fourier series (1.1). The Fourier coefficients are mean zero independent normal variables whose variances depend on the frequencies via an even Schwartz function a on R and large rescaling parameter R. For any open subset U of the torus denote by Z R (U) the number of critical points of F a R in U. We prove that if U is contained in a geodesic ball, then the variance of Z R (U) is asymptotic to const × Rmvol [ U ] as R → ∞. We use this to prove that if m ≥ 2, then as N → ∞, the random measures N − m Z N (−) converge a.s. to an explicit multiple of the volume measure on the flat torus. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Stochastic Processes & Their Applications is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1016/j.spa.2026.104899 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 1 StartPage: N.PAG Subjects: – SubjectFull: Law of large numbers Type: general – SubjectFull: Random functions (Mathematics) Type: general – SubjectFull: Stochastic processes Type: general – SubjectFull: Torus Type: general – SubjectFull: Gaussian distribution Type: general – SubjectFull: Asymptotic distribution Type: general Titles: – TitleFull: A law of large numbers concerning the distribution of critical points of random Fourier series. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: "Brandon" Fu, Qiangang – PersonEntity: Name: NameFull: Nicolaescu, Liviu I. IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 05 Text: May2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 03044149 Numbering: – Type: volume Value: 195 Titles: – TitleFull: Stochastic Processes & Their Applications Type: main |
| ResultId | 1 |
