In 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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