### Abstract

We prove that if I_{k} are disjoint blocks of positive integers and nk are independent random variables on some probability space (Ω, F, P) such that n_{k} is uniformly distributed on I_{k}, then N N^{−}1/^{2} (sin 2πnkx − E(sin 2πnkx)) k=1 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.

Original language | English |
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Pages (from-to) | 141-152 |

Number of pages | 12 |

Journal | Studia Scientiarum Mathematicarum Hungarica |

Volume | 55 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 1 2018 |

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### Keywords

- Central limit theorem
- Random gaps
- Trigonometric sums

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Studia Scientiarum Mathematicarum Hungarica*,

*55*(1), 141-152. https://doi.org/10.1556/012.2018.55.1.1389

**On trigonometric sums with random frequencies.** / Bazarova, Alina; Berkes, I.; Raseta, Marko.

Research output: Contribution to journal › Article

*Studia Scientiarum Mathematicarum Hungarica*, vol. 55, no. 1, pp. 141-152. https://doi.org/10.1556/012.2018.55.1.1389

}

TY - JOUR

T1 - On trigonometric sums with random frequencies

AU - Bazarova, Alina

AU - Berkes, I.

AU - Raseta, Marko

PY - 2018/3/1

Y1 - 2018/3/1

N2 - We prove that if Ik are disjoint blocks of positive integers and nk are independent random variables on some probability space (Ω, F, P) such that nk is uniformly distributed on Ik, then N N−1/2 (sin 2πnkx − E(sin 2πnkx)) k=1 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 nk have continuous uniform distribution on disjoint intervals Ik on the positive axis.

AB - We prove that if Ik are disjoint blocks of positive integers and nk are independent random variables on some probability space (Ω, F, P) such that nk is uniformly distributed on Ik, then N N−1/2 (sin 2πnkx − E(sin 2πnkx)) k=1 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 nk have continuous uniform distribution on disjoint intervals Ik on the positive axis.

KW - Central limit theorem

KW - Random gaps

KW - Trigonometric sums

UR - http://www.scopus.com/inward/record.url?scp=85046934133&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85046934133&partnerID=8YFLogxK

U2 - 10.1556/012.2018.55.1.1389

DO - 10.1556/012.2018.55.1.1389

M3 - Article

VL - 55

SP - 141

EP - 152

JO - Studia Scientiarum Mathematicarum Hungarica

JF - Studia Scientiarum Mathematicarum Hungarica

SN - 0081-6906

IS - 1

ER -