A ``random stationary signal'', more formally known as a Gaussian stationary function, is a random function f:R-->R whose distribution is invariant under real shifts (hence stationary), and whose evaluation at any finite number of points is a centered Gaussian random vector (hence Gaussian). The mathematical study of these random functions goes back at least 75 years, with pioneering works by Kac, Rice and Wiener, who were motivated both by applications in engineering and by analytic questions about ``typical'' behavior in certain classes of functions. Nonetheless, many basic questions about them, such as the fluctuations of their number of zeroes, or the probability of having no zeroes in a large region, remained unanswered for many years. In this talk, we will provide an introduction to Gaussian stationary functions and describe how a spectral perspective, combined with tools from harmonic, real and complex analysis, yields new results about such long-lasting questions.
2018-12-06 14:30 to 15:30
Manchester Building (Hall 2), Hebrew University Jerusalem