## The Annals of Probability

- Ann. Probab.
- Volume 3, Number 5 (1975), 849-858.

### Conditions for Finite Moments of the Number of Zero Crossings for Gaussian Processes

#### Abstract

Let $M_k(0, T)$ denote the $k$th (factorial) moment of the number of zero crossings in time $T$ by a stationary Gaussian process. We present a necessary and sufficient condition for $M_k(0, T)$ to be finite. This condition is then applied to processes whose covariance functions $\rho(t)$ satisfy the local condition. $$\rho(t) = 1 - \frac{t^2}{2} + \frac{C|t|^3}{6} + o|t|^3$$ for $t$ near zero $(C > 0)$. In this case we show all the crossing moments $M_k(0, T)$ are finite. In the course of the proof of this result, we point out an error which vitiates the related work of Piterbarg (1968) and Mirosin (1971, 1973, 1974a, 1974b). We also find a counterexample to Piterbarg's results.

#### Article information

**Source**

Ann. Probab., Volume 3, Number 5 (1975), 849-858.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996271

**Digital Object Identifier**

doi:10.1214/aop/1176996271

**Mathematical Reviews number (MathSciNet)**

MR388515

**Zentralblatt MATH identifier**

0328.60023

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G17: Sample path properties

Secondary: 60G10: Stationary processes 60G15: Gaussian processes

**Keywords**

Zero crossings level crossings Gaussian processes moments factorial moments point processes

#### Citation

Cuzick, Jack. Conditions for Finite Moments of the Number of Zero Crossings for Gaussian Processes. Ann. Probab. 3 (1975), no. 5, 849--858. doi:10.1214/aop/1176996271. https://projecteuclid.org/euclid.aop/1176996271