The easy inequality 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 on a finite time interval is represented as the integral of complex integrand which has a special feature in the neighborhood of zero to make it difficult to computer calculation. In the article for a wide class of correlation functions it is proven inequality to estimate both the top and bottom of the variance in terms of elementary function and without using integrals. Two examples demonstrate the the limits to the effectiveness of this inequality by comparison with earlier established formulas of variance of particular cases of processes for which the variance is also calculated without integrals. In the other three examples the inequality is used for to get the main term of the asymptotic of variance of the number of zeros on the small interval of time. Also in the latter example bounds of the variance of analytical process are estimated on any time intervals. Refs 18.

Keywords:

differentiable Gaussian stationary process, the variance of the number of zeros, correlation function, inequality for the variance, Wong process, asymptotic