We study the persistence probability of a centered stationary Gaussian process on or , that is, its probability to remain positive for a long time. We describe the delicate interplay between this probability and the behavior of the spectral measure of the process near zero and infinity.
"Persistence of Gaussian stationary processes: A spectral perspective." Ann. Probab. 49 (3) 1067 - 1096, May 2021. https://doi.org/10.1214/20-AOP1470