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In one way or another we are all waiting. Even those who seem to have it all are still waiting on something. There may be a young athlete who’s waiting for his wildest dreams to come true. There may be a young lady waiting on an acceptance letter to a university. A married couple may be waiting to find out if they’re giving birth to a boy or a girl. There may be an individual waiting for lab results from his most recent doctor’s visit. We are all in the same boat. Waiting. Waiting for change. Waiting for answers. Waiting for healing. Waiting for things to get better. No matter how good life gets, every living human being will always be in need of something.

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.