The Annals of Mathematical Statistics

The Moments of the Number of Crossings of a Level by a Stationary Normal Process

Harald Cramer and M. R. Leadbetter

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Abstract

In this paper we consider the number $N$ of upcrossings of a level $u$ by a stationary normal process $\xi(t)$ in $0 \leqq t \leqq T$. A formula is obtained for the factorial moment $M_k = \varepsilon\{N(N - 1) \cdots (N - k + 1)\}$ of any desired order $k$. The main condition assumed in the derivation is that $\xi(t)$ have, with probability one, a continuous sample derivative $\xi'(t)$ in the interval $\lbrack 0, T\rbrack$. This condition involves hardly any restriction since an example shows that even a slight relaxation of it causes all moments of order greater than one to become infinite. The moments of the number of downcrossings or total number of crossings can be obtained analogously.

Article information

Source
Ann. Math. Statist., Volume 36, Number 6 (1965), 1656-1663.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177699794

Digital Object Identifier
doi:10.1214/aoms/1177699794

Mathematical Reviews number (MathSciNet)
MR185682

Zentralblatt MATH identifier
0137.35603

JSTOR
links.jstor.org

Citation

Cramer, Harald; Leadbetter, M. R. The Moments of the Number of Crossings of a Level by a Stationary Normal Process. Ann. Math. Statist. 36 (1965), no. 6, 1656--1663. doi:10.1214/aoms/1177699794. https://projecteuclid.org/euclid.aoms/1177699794


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