Abstract:

We study an aspect of the zeros of centered stationary Gaussian

processes (SGP) on R, namely NT, which is the number of zeros in the interval [0,T]. In earlier studies, under varying assumptions on the spectral measure of the SGP, the following results/statistics were obtained for NT: expectation, variance asymptotics, CLT, exponential concentration, overcrowding estimates, and finiteness of moments. We will restrict our attention to SGP with compactly supported spectral measure μ. Let A > 0 be the smallest number such that supp(μ) ⊆ [−A, A]. Stationarity of the process implies that the expectation of NT is proportional to T. Our primary interest is in overcrowding (resp. under crowding) probability, which is the probability of the event that there is an excess (resp. deficit) of zeros in [0,T] compared to the expected number. Comparing a couple of known results, we can conclude that there is 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, this probability is at least of the order of exp(−CηT) for small η, and at most of order exp(−cηT2) for large η. We will also identify the critical η where this transition happens to be ηc = A/π. We also prove a similar result 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.