The Annals of Probability

Local Nondeterminism and the Zeros of Gaussian Processes

Jack Cuzick

Full-text: Open access

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


Export citation

Corrections

  • See Correction: Jack Cuzick. Correction: Local Nondeterminism and the Zeros of Gaussian Processes. Ann. Probab., Volume 15, Number 3 (1987), 1229--1229.