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Bartlett spectrum and mixing properties of infinitely divisible random measures

  • Emmanuel Roy (a1)
Abstract

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.

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Copyright
Corresponding author
Current address: Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 13, UMR 7539, 99 avenue J. B. Clément, F-93430 Villetaneuse, France. Email address: roy@math.univ-paris13.fr
References
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[1] Aaronson, J. (1997). An Introduction to Infinite Ergodic Theory. American Mathematical Society, Providence, RI.
[2] Brémaud, P. and Massoulié, L. (2001). Hawkes branching processes without ancestors. J. Appl. Prob. 38, 122135.
[3] Cornfeld, I. P., Fomin, S. V. and Sinaı˘, Y. G. (1982). Ergodic Theory. Springer, New York.
[4] Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.
[5] Kallenberg, O. (1983). Random Measures. Academic Press, London.
[6] Krengel, U. and Sucheston, L. (1969). On mixing in infinite measure spaces. Z. Wahrscheinlichkeitsth. 13, 150164.
[7] Matthes, K., Kerstan, J. and Mecke, J. (1978). Infinitely Divisible Point Processes. John Wiley, Chichester.
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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
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