On Crossings of Levels and Curves by a Wide Class of Stochastic Processes

Abstract

In this paper, upcrossings, downcrossings and tangencies to levels and curves are discussed within a general framework. The mean number of crossings of a level (or curve) is calculated for a wide class of processes and it is shown that tangencies have probability zero in these cases. This extends results of Ito [1] and Ylvisaker [7] for stationary normal processes, to nonstationary and non normal cases. In particular the corresponding result given by Leadbetter and Cryer [3] for normal, non stationary processes can be slightly improved to apply under minimal conditions. An application is also given for an important non normal process.