Overcrowding estimates for nodal volume of stationary Gaussian processes (SGPs) on R^d

Abstract: We consider centered SGPs on Euclidean spaces R^d and study their nodal volume in [0,T]^d, for T>0. From earlier studies, we know the following statistics for nodal volume of SGPs under varying assumptions on their spectral measures: expectation, variance asymptotics, CLT, exponential concentration (only for d=1), and finiteness of moments.

We study the unlikely event of overcrowding of the nodal set in [0,T]^d; this is the event that the volume of the nodal set in [0,T]^d is much larger than its expected value. Under some mild assumptions on the spectral measure, we obtain estimates for the overcrowding event's probability. We first get overcrowding estimates for the zero count of SGPs on R. In higher dimensions, we consider Crofton's formula, which gives the volume of the nodal set in terms of the number of intersections of the nodal set with all lines in R^d. We discretize this formula to get a more workable version of it and, in a sense, reduce the overcrowding problem in higher dimensions to the one-dimensional case.

probability

Audience: advanced learners

Bangalore Probability Seminar

Series comments: The link to zoom meeting can be found on the seminar's google calendar - www.isibang.ac.in/~d.yogesh/BPS.html