Naomi Feldheim, Stanford University: Let $(X_j)$ be a real Gaussian stationary sequence, that is, a random sequence whose finite marginals have centered multi-variate Gaussian distribution, and $E(X_j X_i) = r(j-i)$. Given the covariance function $r$, what is the behavior of the probability that X_1>0, X_2>0, ... , X_N>0 as N grows to infinity? This simple question goes back at least to the 1950's, when researchers such as Slepian and Rice were developing a theory of noise and signal processing. In the last decade this quantity was found to have strong relations with other models in statistical physics and probability, however, until recently only crude bounds were known. This may be due to the fact that the question is misleading: our recent research shows that persistence is better described by the behavior of the spectral measure, the inverse Fourier transform of $r$, rather than the function $r$ itself. In the talk we formulate a simple description of the persistence probability in terms of the behavior of the spectral measure near zero. The results hold also for Gaussian processes on the real line. Our work simplifies and generalizes bounds by Newell-Rosenblatt (1962), as well as the very recent works by Dembo-Mukherjee (2015), and Krishna-Krishnapur (2016). Joint work with Ohad Feldheim and Shahaf Nitzan.
Location Hill 705