## The Annals of Probability

- Ann. Probab.
- Volume 6, Number 1 (1978), 72-84.

### Local Nondeterminism and the Zeros of Gaussian Processes

#### Abstract

The concept of local nondeterminism introduced by Berman is generalized and applied to divided difference sequences generated by a Gaussian process. The resulting estimates are then used to find simple sufficient conditions for the finiteness of the moments of the number of crossings of level zero. In particular it is shown that under mild regularity conditions very little more is required to make all moments finite when the variance is finite. The results are extended to curves $\xi \in \mathscr{L}_2\lbrack 0, T\rbrack$. Finally examples are given in which the variance is finite but the third moment is infinite.

#### Article information

**Source**

Ann. Probab., Volume 6, Number 1 (1978), 72-84.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176995611

**Digital Object Identifier**

doi:10.1214/aop/1176995611

**Mathematical Reviews number (MathSciNet)**

MR488252

**Zentralblatt MATH identifier**

0374.60051

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G17: Sample path properties

Secondary: 60G15: Gaussian processes 60G25: Prediction theory [See also 62M20]

**Keywords**

Zero crossings curve crossings local nondeterminism Gaussian processes point processes prediction

#### Citation

Cuzick, Jack. Local Nondeterminism and the Zeros of Gaussian Processes. Ann. Probab. 6 (1978), no. 1, 72--84. doi:10.1214/aop/1176995611. https://projecteuclid.org/euclid.aop/1176995611

#### Corrections

- See Correction: Jack Cuzick. Correction: Local Nondeterminism and the Zeros of Gaussian Processes. Ann. Probab., Volume 15, Number 3 (1987), 1229--1229.Project Euclid: euclid.aop/1176992096