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Bartlett spectrum and mixing properties of infinitely divisible random measures

Part of: Stochastic processes Distribution theory - Probability Ergodic theory

Published online by Cambridge University Press:  01 July 2016

Emmanuel Roy
Emmanuel Roy*
Affiliation:
Université Paris 13
*
∗ Current address: Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 13, UMR 7539, 99 avenue J. B. Clément, F-93430 Villetaneuse, France. Email address: [email protected]
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Abstract

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We prove that the Bartlett spectrum of a stationary, infinitely divisible (ID) random measure determines ergodicity, weak mixing, and mixing. In this context, the Bartlett spectrum plays the same role as the spectral measure of a stationary Gaussian process.

Keywords

Infinitely divisible random measureinfinitely divisible point processCox processBartlett spectrumergodic theory

MSC classification

Primary: 60G57: Random measures 60G55: Point processes
Secondary: 37A05: Measure-preserving transformations 60E07: Infinitely divisible distributions; stable distributions
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 39 , Issue 4 , December 2007 , pp. 893 - 897
Copyright
Copyright © Applied Probability Trust 2007 

References

Aaronson, J. (1997). An Introduction to Infinite Ergodic Theory. American Mathematical Society, Providence, RI.Google Scholar
Brémaud, P. and Massoulié, L. (2001). Hawkes branching processes without ancestors. J. Appl. Prob. 38, 122–135.Google Scholar
Cornfeld, I. P., Fomin, S. V. and Sinaı˘, Y. G. (1982). Ergodic Theory. Springer, New York.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Kallenberg, O. (1983). Random Measures. Academic Press, London.Google Scholar
Krengel, U. and Sucheston, L. (1969). On mixing in infinite measure spaces. Z. Wahrscheinlichkeitsth. 13, 150–164.CrossRefGoogle Scholar
Matthes, K., Kerstan, J. and Mecke, J. (1978). Infinitely Divisible Point Processes. John Wiley, Chichester.Google Scholar