Abstract
The equivalence of ergodicity and weak mixing for general infinitely divisible processes is proven. Using this result and [9], simple conditions for ergodicity of infinitely divisible processes are derived. The notion of codifference for infinitely divisible processes is investigated, it plays the crucial role in the proofs but it may be also of independent interest.
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REFERENCES
Cambanis, S., Podgórski, K., and Weron, A. (1995). Chaotic behavior of infinitely divisible processes, Studia Math. 115, 109–127.
Cornfeld, I. P., Fomin, S. V., and Sinai, Ya. G. (1982). Ergodic Theory, Springer-Verlag.
Gross, A., and Robertson, J. B. (1993). Ergodic properties of random measures on stationary sequences of sets, Stoch. Proc. Appl. 46, 249–265.
Kokoszka, P., and Podgórski, K. (1992). Ergodicity and weak mixing of semistable processes, Prob. and Math. Stat. 13, 239–244.
Koopman, B. O., and v. Neumann, J. (1932). Dynamical systems of continuous spectra, Proc. Nat. Acad. Sci. U.S.A. 18, 255–263.
Lukacs, E. (1980). Characteristic Functions, Hafner.
Petersen, K. (1983). Ergodic Theory, Cambridge University Press.
Podgórski, K. (1992). A note on ergodic symmetric stable processes, Stoch. Proc. Appl. 43, 355–362.
Rosiński, J., and Zak, T. (1996). Simple conditions for mixing of infinitely divisible processes, Stoch. Proc. Appl. 61, 277–288.
Samorodnitsky, G., and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes, Chapman and Hall.
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Rosiński, J., Żak, T. The Equivalence of Ergodicity and Weak Mixing for Infinitely Divisible Processes. Journal of Theoretical Probability 10, 73–86 (1997). https://doi.org/10.1023/A:1022690230759
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DOI: https://doi.org/10.1023/A:1022690230759