## Random Fourier Series with Applications to Harmonic AnalysisIn this book the authors give the first necessary and sufficient conditions for the uniform convergence a.s. of random Fourier series on locally compact Abelian groups and on compact non-Abelian groups. They also obtain many related results. For example, whenever a random Fourier series converges uniformly a.s. it also satisfies the central limit theorem. The methods developed are used to study some questions in harmonic analysis that are not intrinsically random. For example, a new characterization of Sidon sets is derived. The major results depend heavily on the Dudley-Fernique necessary and sufficient condition for the continuity of stationary Gaussian processes and on recent work on sums of independent Banach space valued random variables. It is noteworthy that the proofs for the Abelian case immediately extend to the non-Abelian case once the proper definition of random Fourier series is made. In doing this the authors obtain new results on sums of independent random matrices with elements in a Banach space. The final chapter of the book suggests several directions for further research. |

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### Contents

INTRODUCTION | 3 |

PRELIMINARIES | 16 |

2 A Jensen type inequality for the nondecreasing rearrangement of nonnegative stochastic processes | 19 |

3 Continuity of Gaussian and subGaussian processes | 24 |

4 Sums of Banach space valued random variables | 40 |

RANDOM FOURIER SERIES ON LOCALLY COMPACT ABELIAN GROUPS | 51 |

2 Random Fourier series on the real line | 60 |

3 Random Fourier series on compact Abelian groups | 63 |

3 Continuity of random Fourier series | 93 |

APPLICATIONS TO HARMONIC ANALYSIS | 105 |

2 Applications to Sidon sets | 118 |

ADDITIONAL RESULTS AND COMMENTS | 122 |

2 Almost sure almost periodicity | 134 |

3 On left and right almost sure continuity | 138 |

4 Generalizations | 140 |

REFERENCES | 144 |

THE CENTRAL LIMIT THEOREM AND RELATED QUESTIONS | 65 |

RANDOM FOURIER SERIES ON COMPACT NONABELIAN GROUPS | 74 |

2 Random series with coefficients in a Banach space | 81 |