Consider a real Gaussian stationary process, either on Z or on R. That is, a stochastic process, invariant under translations, whose finite marginals are centered multi-variate Gaussians. The persistence of such a process on [0,N] is the probability that it remains positive throughout this interval. The relation between the decay of the persistence as N tends to infinity and the covariance function of the process has been investigated since the 1950s with motivations stemming from probability, engineering and mathematical physics. Nonetheless, until recently, good estimates were known only for particular cases, or when the covariance kernel of the process is either non-negative or summable. In the talk we discuss a new spectral point of view on persistence which greatly simplifies its analysis. This is then used to analyze the qualitative behavior of the persistence probability in a very general setting. Based on joint work with Ohad Feldheim and Shahaf Nitzan.
Tue, 20/06/2017 - 14:00 to 15:00