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Probability seminar: Ohad Feldheim (HUJI) - A phase transition in zero count probability for Stationary Gaussian Processes | Einstein Institute of Mathematics

Probability seminar: Ohad Feldheim (HUJI) - A phase transition in zero count probability for Stationary Gaussian Processes

Date: 
Mon, 14/11/202214:00-15:00
Location: 
שפרינצק 29

Consider a centered real stationary Gaussian process f(t), that is, a random real function with multinormal marginals whose distribution is invariant under translations. This is the most commonly used model for random noise (random signals, ocean surface fluctuations, etc’). The expected number of zeroes of such a process in an interval [0,T] is bT, where b can be computed by the celebrated Kac-Rice formula.  Here we study the probability of a significant deviation of this random variable, known as overcrowding and undercrowding of zeroes, given by having at least (b+d)T zeroes, or at most (b-d)T respectively. We show that when the support of the spectrum of the process is separated from infinity (respectively from zero) there is a sharp phase transition in d in the overcrowding probability (respectively in undercrowding), at a value proportional to the edge of the support of the spectrum. At this point the probability of the rare event in question changes abruptly from exponential to sub-Gaussian. 

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 GSPs is assumed.

Joint work with Naomi Feldheim & Lakshmi Priya.