# The Billard Theorem for Multiple Random Fourier Series

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## Abstract

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.

## Keywords

Billard Theorem Random Fourier series Multiple Fourier series Random phase Random fields## Mathematics Subject Classification

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

### Acknowledgments

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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