Convergence of Random Fourier Series

This paper will study Fourier Series with random coefficients. We begin with an introduction to Fourier series on the torus and give some of the most important results. We then give some important results from probability theory, and build on these to prove a variety of theorems that deal with the convergence or divergence of general random series. In the final section, the focus is placed on random Fourier series, and we combine results from the previous sections to prove our main theorem. The main result of this paper gives a simple condition for the almost-everywhere convergence or divergence of a random trigonometric series, and proves that divergence implies that the coefficients cannot be the Fourier series of any function.