MWI representation of the number of curve-crossings by a differentiable Gaussian process, with applications. (English) Zbl 0819.60036

Slud, Eric V.

Ann. Probab. 22, No. 3, 1355-1380 (1994).

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.

Reviewer: M.I.Yadrenko (Kiev)

MSC:

60G15	Gaussian processes
60F05	Central limit and other weak theorems
60G35	Signal detection and filtering (aspects of stochastic processes)

Keywords:

Rice’s formula; multiple Wiener integral; central and noncentral limit theorems