Jeremiah Buckley

An asymptotic formula for the variance of the number of zeroes of a stationary Gaussian process

The zeroes of a stationary Gaussian process on the real line are a classical object, and the mean number of zeroes is given by the famous Kac-Rice formula. A formula in a similar spirit, due to Cram \'{e}r-Leadbetter, computes the variance exactly. Unfortunately this expression is not very accessible, and it is difficult to get a good estimate for the size of the variance. We will discuss an approximate formula, that computes the asymptotic growth of the variance in a long interval, under mild mixing conditions. In particular, we give a linear lower bound for any non-degenerate process. We will also discuss atoms in the spectral measure, and the emergence of a `special frequency’. Joint work with Eran Assaf and Naomi Feldheim.