Strong approximation of lacunary series with random gaps

Alina Bazarova; Istvan Berkes; Marko Raseta

doi:10.1007/s00605-017-1059-5

Strong approximation of lacunary series with random gaps

Alina Bazarova, Istvan Berkes^*, Marko Raseta

^*Corresponding author for this work

Institute of Statistics (5060)

Research output: Contribution to journal › Article › peer-review

Abstract

We investigate the asymptotic behavior of sums ∑^N_k=1f(n_kx), where f is a mean zero, smooth periodic function on ℝ and (n_k)_k≥1 is a random sequence such that the gaps n_k₊₁- n_k are i.i.d. Our result shows that, in contrast to the classical Salem–Zygmund theory, the almost sure behavior of lacunary series with random gaps can be described very precisely without any assumption on the size of the gaps.

Original language	English
Pages (from-to)	393-406
Number of pages	14
Journal	Monatshefte fur Mathematik
Volume	186
Issue number	3
DOIs	https://doi.org/10.1007/s00605-017-1059-5
Publication status	Published - 26 May 2017

Keywords

Lacunary series
Random indices
Wiener approximation

ASJC Scopus subject areas

General Mathematics

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10.1007/s00605-017-1059-5

strong_approximation_of_lacunary_series_with_random_gapsAccepted author manuscript, 154 KBLicence: CC BY 4.0

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title = "Strong approximation of lacunary series with random gaps",

abstract = "We investigate the asymptotic behavior of sums ∑Nk=1f(nkx), where f is a mean zero, smooth periodic function on ℝ and (nk)k≥1 is a random sequence such that the gaps nk+1- nk are i.i.d. Our result shows that, in contrast to the classical Salem–Zygmund theory, the almost sure behavior of lacunary series with random gaps can be described very precisely without any assumption on the size of the gaps.",

keywords = "Lacunary series, Random indices, Wiener approximation, Lacunary series, Random indices, Wiener approximation",

author = "Alina Bazarova and Istvan Berkes and Marko Raseta",

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T1 - Strong approximation of lacunary series with random gaps

AU - Bazarova, Alina

AU - Berkes, Istvan

AU - Raseta, Marko

PY - 2017/5/26

Y1 - 2017/5/26

N2 - We investigate the asymptotic behavior of sums ∑Nk=1f(nkx), where f is a mean zero, smooth periodic function on ℝ and (nk)k≥1 is a random sequence such that the gaps nk+1- nk are i.i.d. Our result shows that, in contrast to the classical Salem–Zygmund theory, the almost sure behavior of lacunary series with random gaps can be described very precisely without any assumption on the size of the gaps.

AB - We investigate the asymptotic behavior of sums ∑Nk=1f(nkx), where f is a mean zero, smooth periodic function on ℝ and (nk)k≥1 is a random sequence such that the gaps nk+1- nk are i.i.d. Our result shows that, in contrast to the classical Salem–Zygmund theory, the almost sure behavior of lacunary series with random gaps can be described very precisely without any assumption on the size of the gaps.

KW - Lacunary series

KW - Random indices

KW - Wiener approximation

KW - Lacunary series

KW - Random indices

KW - Wiener approximation

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DO - 10.1007/s00605-017-1059-5

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SP - 393

EP - 406

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

IS - 3

ER -

Strong approximation of lacunary series with random gaps

Abstract

Keywords

ASJC Scopus subject areas

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