## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 37, Number 1 (1966), 260-267.

### 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.

#### Article information

**Source**

Ann. Math. Statist., Volume 37, Number 1 (1966), 260-267.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177699615

**Digital Object Identifier**

doi:10.1214/aoms/1177699615

**Mathematical Reviews number (MathSciNet)**

MR208667

**Zentralblatt MATH identifier**

0141.14906

**JSTOR**

links.jstor.org

#### Citation

Leadbetter, M. R. On Crossings of Levels and Curves by a Wide Class of Stochastic Processes. Ann. Math. Statist. 37 (1966), no. 1, 260--267. doi:10.1214/aoms/1177699615. https://projecteuclid.org/euclid.aoms/1177699615