## The Annals of Probability

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

Jack Cuzick

#### 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

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