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

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.