Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T07:34:21.171Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Wolfgang Stadje
Affiliation:
Department of Mathematics, and Computer Science, University of Osnabrück, 49069 Osnabrück, 49069 Osnabrück, Germany e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Bertoin, J., Lévy processes (Cambridge University Press, Cambridge, 1996).Google Scholar
[2]Chung, K. L., A course in probability theory (Harcourt, Brace & World, New York, 1968).Google Scholar
[3]Doob, J. L., Stochastic processes (Wiley, New York, 1953).Google Scholar
[4]Fristedt, B., ‘Sample functions of stochastic processes with stationary independent increments’, in: Advances in probability and related topics, Vol. 3 (eds. Ney, P. and port, S.) (Dekker, New York, 1974) pp. 241–396.Google Scholar
[5]Hartman, P. and Wintner, A., ‘On the infinitesimal generator of integral convolutions’, Amer. J. Math 64 (1942), 273–298.CrossRefGoogle Scholar
[6]Stadje, W., ‘Two ergodic sample-path properties of the Poisson process’, J. Theor. Probab. 11 (1998), 197–208.CrossRefGoogle Scholar
[7]Stidham, S. and El-Taha, M., ‘Sample-path techniques in queueing theory,’ in: Advances in queueing (ed. Dshalalow, J. W.) CRC Press, Boca Raton, (1998) pp. 119–166.Google Scholar
[8]Stroock, D. W., Probability theory: an analytic view (Cambridge University Press, Cambridge, 1992).Google Scholar