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The envelope of an oscillatory process and its upcrossings

Published online by Cambridge University Press:  01 July 2016

A. M. Hasofer  and
P. Petocz
A. M. Hasofer
Affiliation:
University of New South Wales
P. Petocz
Affiliation:
University of New South Wales
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Extract

Let X(t) be a real-valued stochastic process with sample functions which are almost surely continuously differentiable. We say that X(t) has an upcrossing at level u at time t if X(t) = u, X′(t) > 0.

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 4 , December 1978 , pp. 711 - 716
Copyright
Copyright © Applied Probability Trust 1978 

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References

[1] Cramér, H. and Leadbetter, M. R. (1967) Stationary and Related Stochastic Processes. Wiley, New York.Google Scholar
[2] Leadbetter, M. R. (1966) On crossings of levels and curves by a wide class of stochastic processes. Ann. Math. Statist. 37, 260–267.Google Scholar
[3] Priestley, M. B. (1965) Evolutionary spectra and non-stationary processes. J. R. Statist. Soc. B 27, 204–229.Google Scholar