On trigonometric sums with random frequencies

Alina Bazarova; István Berkes; Marko Raseta

doi:10.1556/012.2018.55.1.1389

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Studia Scientiarum Mathematicarum Hungarica

Volume/Issue:: Volume 55: Issue 1

On trigonometric sums with random frequencies

Authors:

Alina Bazarova

Alina Bazarova University of Birmingham, Institute of Cancer and Genomic Sciences, Center for Computational Biology

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a.bazarova@bham.ac.uk

István Berkes

István Berkes Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13{15, 1053 Budapest, Hungary

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berkes.istvan@renyi.mta.hu

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Marko Raseta

Marko Raseta University of Keele, Research Institute for Primary Care and Health Sciences and Research Institute for Applied Clinical Sciences

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m.raseta@keele.ac.uk

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Pages:: 141–152

Online Publication Date:: Mar 2018

Publication Date:: 01 Mar 2018

Article Category:: Journal Article

DOI:: https://doi.org/10.1556/012.2018.55.1.1389

Keywords:: Trigonometric sums; random gaps; central limit theorem

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We prove that if I_k are disjoint blocks of positive integers and n_k are independent random variables on some probability space (Ω,F,P) such that n_k is uniformly distributed on I_k, then $N^{- 1 / 2} \sum_{k = 1}^{N} (\sin 2 π n_{k} x - E (\sin 2 π n_{k} x))$ has, with P-probability 1, a mixed Gaussian limit distribution relative to the probability space ((0, 1),B, λ), where B is the Borel σ-algebra and λ is the Lebesgue measure. We also investigate the case when n_k have continuous uniform distribution on disjoint intervals I_k on the positive axis.

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Studia Scientiarum Mathematicarum Hungarica

Print ISSN:: 0081-6906

Online ISSN:: 1588-2896

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Gábor SIMONYI (Rényi Institute of Mathematics)
András STIPSICZ (Rényi Institute of Mathematics)
Géza TÓTH (Rényi Institute of Mathematics)

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Gábor SÁGI (Rényi Institute of Mathematics)

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Imre BÁRÁNY (Rényi Institute of Mathematics)
Károly BÖRÖCZKY (Rényi Institute of Mathematics and Central European University)
Péter CSIKVÁRI (ELTE, Budapest)
Joshua GREENE (Boston College)
Penny HAXELL (University of Waterloo)
Andreas HOLMSEN (Korea Advanced Institute of Science and Technology)
Ron HOLZMAN (Technion, Haifa)
Satoru IWATA (University of Tokyo)
Tibor JORDÁN (ELTE, Budapest)
Roy MESHULAM (Technion, Haifa)
Frédéric MEUNIER (École des Ponts ParisTech)
Márton NASZÓDI (ELTE, Budapest)
Eran NEVO (Hebrew University of Jerusalem)
János PACH (Rényi Institute of Mathematics)
Péter Pál PACH (BME, Budapest)
Andrew SUK (University of California, San Diego)
Zoltán SZABÓ (Princeton University)
Martin TANCER (Charles University, Prague)
Gábor TARDOS (Rényi Institute of Mathematics)
Paul WOLLAN (University of Rome "La Sapienza")

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Studia Scientiarum Mathematicarum Hungarica
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