MWI REPRESENTATION OF THE NUMBER OF CURVE-CROSSINGS BY A DIFFERENTIABLE GAUSSIAN PROCESS, WITH APPLICATIONS

Let X = (X(t),t greater than or equal to 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 X on the bounded interval [0, T], in a multiple Wiener-It (o) over cap integral expansion. This continues work of the author in which the result was given for psi equivalent to 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.