Zeros and Winding of Stationary Gaussian Processes: Sharp Transitions in Probability Decay Rates

Consider a centered stationary Gaussian process, that is, a random function (either real or complex) with multi-normal marginals, whose distribution is invariant under translations. This is the most common model for random noise (random signals, ocean surface fluctuations, etc’). In the real case we consider the number of zeroes, while in the complex case we consider the winding of the process around zero — both on a large interval [0,T]. The expectation of these is bT, and b can be computed by the celebrated Kac-Rice formula.

Here we study the probability of a significant deviation of these random variables from their expectation. In the real case this is given by having at least (b+d)T zeroes (d-overcrowding), or at most (b-d)T zeroes (d-undercrowding); the complex analogues are overwinding & underwinding. We show a sharp transition in the decay rate for the probabilities of these events as a function of d, pertaining to the spectral support of the process. For example, when the support of the spectrum of a real process is [-a,a], the d-overcrowding probability decays exponentially for d<a*pi, and sub-Gaussianly for d>a*pi.

The methods used to show these results involve tools from the theory of Gaussian processes and complex analysis. The topic will be fully introduced and no prior familiarity with stationary Gaussian processes is assumed.

Based on joint works with Naomi Feldheim & Lakshmi Priya, and with Itamar Zangvil.