Journal of Fourier Analysis and Applications

, Volume 23, Issue 1, pp 207–228 | Cite as

The Billard Theorem for Multiple Random Fourier Series

  • Samuel Ronsin
  • Hermine Biermé
  • Lionel MoisanEmail author


We propose a generalization of a classical result on random Fourier series, namely the Billard Theorem, for random Fourier series over the d-dimensional torus. We provide an investigation of the independence with respect to a choice of a sequence of partial sums (or method of summation). We also study some probabilistic properties of the resulting sum field such as stationarity and characteristics of the marginal distribution.


Billard Theorem Random Fourier series Multiple Fourier series Random phase Random fields 

Mathematics Subject Classification

Primary 42B05 60G60 60G17 Secondary 42B08 60G50 



We thank the anonymous referee for his useful comments, which allowed us to improve the quality of the manuscript. This work was supported by the French National Research Agency grant “MATAIM” ANR-09-BLAN-0029-01.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.MAP5, UMR CNRS 8145, Université Paris Descartes, Sorbonne Paris CitéParisFrance
  2. 2.LMA, UMR CNRS 7348, Université de PoitiersChasseneuilFrance

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