## The Annals of Probability

- Ann. Probab.
- Volume 22, Number 3 (1994), 1355-1380.

### 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.

#### Article information

**Source**

Ann. Probab., Volume 22, Number 3 (1994), 1355-1380.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176988606

**Digital Object Identifier**

doi:10.1214/aop/1176988606

**Mathematical Reviews number (MathSciNet)**

MR1303648

**Zentralblatt MATH identifier**

0819.60036

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G15: Gaussian processes

Secondary: 60F05: Central limit and other weak theorems 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx]

**Keywords**

Asymptotic variance central and noncentral limit theorems Hermite polynomials mixed spectrum multiple Wiener-Ito integral Rice's formula spectral representation

#### Citation

Slud, Eric V. MWI Representation of the Number of Curve-Crossings by a Differentiable Gaussian Process, with Applications. Ann. Probab. 22 (1994), no. 3, 1355--1380. doi:10.1214/aop/1176988606. https://projecteuclid.org/euclid.aop/1176988606