A sharp transition in zero overcrowding and undercrowding probabilities for Stationary Gaussian Processes

Abstract

We study the probability that a real stationary Gaussian process has at least $\eta T$ zeros in $[0,T]$ (overcrowding), or at most this number (undercrowding). We show that if the spectral measure of the process is supported on $\pm [B,A]$, overcrowding probability transitions from exponential decay to Gaussian decay at $\eta =\frac{A}{\pi }$, while undercrowding probability undergoes the reverse transition at $\eta =\frac{B}{\pi }$.

arXiv comment17 pages