On bounds for the variance of the number of zeros of differentiable Gaussian stationary process

Abstract

It is known that the variance of the number of zeros of differentiable Gaussian stationary process with the continuous component in the spectrum of a correlation function can be represented by the integral of a sophisticated integrand. Previously, author obtained both upper and lower bounds of this integral under certain conditions in analytical form. In the article, these conditions are checked for several classes of processes, which include part of the first-order Markov processes and two classes of analytic processes. Furthermore, it is shown that the variance of the number of zeros can be obtained with these bounds for process with the correlation function, which has no continuous component in the spectrum. For certain analytic process it is possible to write the variance of the number of zeros in elementary functions (previously such formulas were known for only two processes). Refs 12.

Keywords:

differentiable Gaussian stationary process, inequalities for the variance of the number of zeros, correlation function, cosine process, analytic process, the variance of the number of zeros in elementary functions