MWI Representation of the Number of Curve-Crossings by a Differentiable Gaussian Process, with Applications

Abstract

Let $\mathbf{X} = (X_t, t \geq 0)$ be a stationary Gaussian process with zero mean, continuous spectral distribution and twice-differentiable correlation function. An explicit representation is given for the number $N_\psi(T)$ of crossings of a $C^1$ curve $\psi$ by $\mathbf{X}$ on the bounded interval $\lbrack 0, T\rbrack$, in a multiple Wiener-Ito integral expansion. This continues work of the author in which the result was given for $\psi \equiv 0$. The representation is applied to prove new central and noncentral limit theorems for numbers of crossings of constant levels, and some consequences for asymptotic variances are given in mixed-spectrum settings.