Zero Crossings

In this essay, we presuppose basic knowledge of stochastic processes [1]. Let {Xt : t ≥ 0} be a zero mean, unit variance, stationary Gaussian process with twice differentiable correlation function r(|s − t|) = Cov(Xs,Xt). We wish to study the distribution of lengths of intervals between zeroes of Xt. There are two cases: the first in which r(τ) is analytic (implying differentiability up to all orders) and the second in which the third derivative of r(τ) possesses a jump discontinuity at τ = 0. Define fm(τ) to be the probability density associated with the interval length τ between an arbitrary zero t0 and the (m + 1) st later zero tm+1. In particular, f0(τ) is the probability density of differences between successive zeroes t0 and t1. We will focus on the limiting behavior of fm(τ) as τ → 0. When r(τ) is analytic, it is clear that