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On upcrossing inequalities for subadditive superstationary processes

Published online by Cambridge University Press:  19 September 2008

Michael Krawczak
Michael Krawczak
Affiliation:
Institut für Mathematische Stochastik, Lotzestr. 13, D-3400 Göttingen, West Germany
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Abstract

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Bishop [2]has given a proof of Birkhoff's ergodic theorem by establishing upcrossing inequalities similar to those of Doob. Such inequalities can be considered as quantitative improvements of convergence theorems: while convergence a.e. means that the number of upcrossings of any interval is a.e. finite, they assert integrability and prove bounds for the integrals. The main point of this paper is to prove upcrossing inequalities for the class of subadditive superstationary processes introduced by Abid [1] as a common generalization of Kingman's [5] subadditive stationary processes and Krengel's [6] superstationary processes. We make use of ideas of Smeltzer [7] who handled the subadditive stationary discrete parameter case in his unpublished thesis. In the continuous parameter case our upcrossing inequality requires more restrictive conditions than the corresponding convergence theorem, due to Hachem [3]. We actually show by example that the number of upcrossings need not be integrable under the assumptions of Hachem even for additive stationary processes.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 5 , Issue 3 , September 1985 , pp. 409 - 416
Copyright
Copyright © Cambridge University Press 1985

References

REFERENCES

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