We study an aspect of the zeros of centered stationary Gaussianprocesses (SGP) on R, namely NT, which is the number of zeros in theinterval [0,T]. In earlier studies, under varying assumptions on thespectral measure of the SGP, the following results/statistics wereobtained for NT: expectation, variance asymptotics, CLT, exponentialconcentration, overcrowding estimates, and finiteness of moments.We will restrict our attention to SGP with compactly supported spectralmeasure μ. Let A > 0 be the smallest number such that supp(μ) ⊆ [−A, A].Stationarity of the process implies that the expectation of NT isproportional to T. Our primary interest is in overcrowding (resp. undercrowding) probability, which is the probability of the event that thereis an excess (resp. deficit) of zeros in [0,T] compared to the expectednumber. Comparing a couple of known results, we can conclude that thereis a change in the behaviour of the probability P(NT ≥ ηT), as η varies.We show that there is indeed a sharp transition. That is, thisprobability is at least of the order of exp(−CηT) for small η, and atmost of order exp(−cηT2) for large η. We will also identify the criticalη where this transition happens to be ηc = A/π. We also prove a similarresult for under crowding probability when supp(μ) has a gap at the origin.This talk is based on a joint work with Naomi Feldheim & Ohad Feldheim.
A sharp transition in the zero count of stationary Gaussian processes
Abstract:
