Random Fourier Series and Trigonometric Sums, I

Abstract

In Chapter 3 we investigate Gaussian processes in “the stationary case,” where e.g. the underlying space is a compact group and the distance is translation invariant. This is relevant to the study of random Fourier series, the basic example of which is X t =∑ k≥1 ξ k exp(2P iikt), where t∈[0,1] and the r.v.s ξ k are independent. The fundamental case where ξ k =a k g k for numbers a k and independent Gaussian r.v.s (g k ) is of great historical importance. We prove some of the classical Marcus-Pisier results, which provide a complete solution in this case, and are also quite satisfactory in the more general case when the random coefficients (ξ k ) are square-integrable. We also explain a result of X. Fernique on vector-valued random Fourier series, which had an important part in the genesis of this book.