MWI representation of the number of curve-crossings by a differentiable Gaussian process, with applications (Q1345603)

Let \(X_ t\), \(t \geq 0\), be a stationary Gaussian process with zero mean and 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 \(X_ t\) on the bounded interval \([0,T]\) in the form of a multiple Wiener-Itô integral expansion. The representation is applied to prove new central and non- central limit theorems for numbers of crossings of constant level.