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Ergodic path properties of processes with stationary increments

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  09 April 2009

Offer Kella  and
Wolfgang Stadje
Offer Kella
Affiliation:
Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel e-mail: offer.kella@huji.ac.il
Wolfgang Stadje
Affiliation:
Department of Mathematics, and Computer Science, University of Osnabrück, 49069 Osnabrück, 49069 Osnabrück, Germany e-mail: wolfgang@mathematik.uni-osnabrueck.de
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Abstract

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For a real-valued ergodic process X with strictly stationary increments satisfying some measurability and continuity assumptions it is proved that the long-run ‘average behaviour’ of all its increments over finite intervals replicates the distribution of the corresponding increments of X in a strong sense. Moreover, every Lévy process has a version that possesses this ergodic path property.

MSC classification

Secondary: 60G17: Sample path properties 60G51: Processes with independent increments; L'evy processes 60F15: Strong theorems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 2 , April 2002 , pp. 199 - 208
Copyright
Copyright © Australian Mathematical Society 2002

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