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.
Publié le : 1994-07-14
Classification:
Asymptotic variance,
central and noncentral limit theorems,
Hermite polynomials,
mixed spectrum,
multiple Wiener-Ito integral,
Rice's formula,
spectral representation,
60G15,
60F05,
60G35
@article{1176988606,
author = {Slud, Eric V.},
title = {MWI Representation of the Number of Curve-Crossings by a Differentiable Gaussian Process, with Applications},
journal = {Ann. Probab.},
volume = {22},
number = {4},
year = {1994},
pages = { 1355-1380},
language = {en},
url = {http://dml.mathdoc.fr/item/1176988606}
}
Slud, Eric V. MWI Representation of the Number of Curve-Crossings by a Differentiable Gaussian Process, with Applications. Ann. Probab., Tome 22 (1994) no. 4, pp. 1355-1380. http://gdmltest.u-ga.fr/item/1176988606/