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Simple conditions for mixing of infinitely divisible processes

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  • Rosinski, Jan
  • Zak, Tomasz

Abstract

Let (Xt)t[epsilon]T be a real-valued, stationary, infinitely divisible stochastic process. We show that (Xt)t[epsilon]T is mixing if and only if Eei(Xt - X0) --> EeiX02, provided the Lévy measure of X0 has no atoms in 2[pi]Z. We also show that if (Xt)t[epsilon]T is given by a stochastic integral with respect to an infinitely divisible measure then the mixing of (Xt)t[epsilon]T is equivalent to the essential disjointness of the supports of the representing functions.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:61:y:1996:i:2:p:277-288
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    References listed on IDEAS

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    1. Gross, Aaron, 1994. "Some mixing conditions for stationary symmetric stable stochastic processes," Stochastic Processes and their Applications, Elsevier, vol. 51(2), pages 277-295, July.
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    Cited by:

    1. 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.
    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. Magdziarz, Marcin, 2009. "Correlation cascades, ergodic properties and long memory of infinitely divisible processes," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3416-3434, October.
    4. Ibragimov, Ildar & Kabluchko, Zakhar & Lifshits, Mikhail, 2019. "Some extensions of linear approximation and prediction problems for stationary processes," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2758-2782.
    5. 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.
    6. Kabluchko, Zakhar & Schlather, Martin, 2010. "Ergodic properties of max-infinitely divisible processes," Stochastic Processes and their Applications, Elsevier, vol. 120(3), pages 281-295, March.

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