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The Equivalence of Ergodicity and Weak Mixing for Infinitely Divisible Processes

Author

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  • Jan Rosiński

    (University of Tennessee)

  • Tomasz Żak

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.

Suggested Citation

  • Jan Rosiński & Tomasz Żak, 1997. "The Equivalence of Ergodicity and Weak Mixing for Infinitely Divisible Processes," Journal of Theoretical Probability, Springer, vol. 10(1), pages 73-86, January.
    • Handle: RePEc:spr:jotpro:v:10:y:1997:i:1:d:10.1023_a:1022690230759
    DOI: 10.1023/A:1022690230759
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    References listed on IDEAS

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    1. Rosinski, Jan & Zak, Tomasz, 1996. "Simple conditions for mixing of infinitely divisible processes," Stochastic Processes and their Applications, Elsevier, vol. 61(2), pages 277-288, February.
    2. Gross, Aaron & Robertson, James B., 1993. "Ergodic properties of random measures on stationary sequences of sets," Stochastic Processes and their Applications, Elsevier, vol. 46(2), pages 249-265, June.
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    Cited by:

    1. Tomasz Bojdecki & Luis G. Gorostiza & Anna Talarczyk, 2015. "From intersection local time to the Rosenblatt process," Journal of Theoretical Probability, Springer, vol. 28(3), pages 1227-1249, September.
    2. Zakhar Kabluchko & Mikhail Lifshits, 2017. "Least Energy Approximation for Processes with Stationary Increments," Journal of Theoretical Probability, Springer, vol. 30(1), pages 268-296, March.
    3. Valentin Courgeau & Almut E. D. Veraart, 2022. "Likelihood theory for the graph Ornstein-Uhlenbeck process," Statistical Inference for Stochastic Processes, Springer, vol. 25(2), pages 227-260, July.
    4. Riccardo Passeggeri & Almut E. D. Veraart, 2019. "Mixing Properties of Multivariate Infinitely Divisible Random Fields," Journal of Theoretical Probability, Springer, vol. 32(4), pages 1845-1879, December.
    5. Karling, Maicon J. & Lopes, Sílvia R.C. & de Souza, Roberto M., 2023. "Multivariate α-stable distributions: VAR(1) processes, measures of dependence and their estimations," Journal of Multivariate Analysis, Elsevier, vol. 195(C).

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