Strong approximation of lacunary series with random gaps

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Abstract

We investigate the asymptotic behavior of sums \(\sum _{k=1}^N f(n_kx)\), where f is a mean zero, smooth periodic function on \(\mathbb {R}\) and \((n_k)_{k\ge 1}\) 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.

Keywords

Lacunary series Random indices Wiener approximation 

Mathematics Subject Classification

Primary 60F17 42A55 42A61 

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Copyright information

© Springer-Verlag Wien 2017

Authors and Affiliations

  1. 1.Systems Biology CentreUniversity of WarwickCoventryUK
  2. 2.Alfréd Rényi Institute of MathematicsBudapestHungary
  3. 3.Research Institute for Primary Care and Health SciencesKeele UniversityKeeleUK
  4. 4.Research Institute for Applied Clinical SciencesKeele UniversityKeeleUK

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