Point Processes of Upcrossings and Local Maxima

Abstract

In the limit theory for the maximum of a stationary normal process ξ(t), as developed in Chapter 8, substantial use was made of upcrossings, and of the obvious fact that the maximum exceeds u if there is at least one upcrossing of the level u. However, the upcrossings have an interest in their own right, and as we shall see here, they also contain considerable information about the local structure of the process. This chapter is devoted to the asymptotic Poisson character of the point process of upcrossings of increasingly high levels, and of the point process formed by the local maxima of the process.