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The Dichotomy of Recurrence and Transience of Semi-Lévy Processes

Author

Listed:
  • Makoto Maejima

    (Keio University)

  • Taisuke Takamune

    (Keio University)

  • Yohei Ueda

    (Keio University)

Abstract

A semi-Lévy process is an additive process with periodically stationary increments. In particular, it is a generalization of a Lévy process. The dichotomy of recurrence and transience of Lévy processes is well known, but this is not necessarily true for general additive processes. In this paper, we prove the recurrence and transience dichotomy of semi-Lévy processes. For the proof, we introduce a concept of semi-random walk and discuss its recurrence and transience properties. An example of semi-Lévy process constructed from two independent Lévy processes is investigated. Finally, we prove the laws of large numbers for semi-Lévy processes.

Suggested Citation

  • Makoto Maejima & Taisuke Takamune & Yohei Ueda, 2014. "The Dichotomy of Recurrence and Transience of Semi-Lévy Processes," Journal of Theoretical Probability, Springer, vol. 27(3), pages 982-996, September.
    • Handle: RePEc:spr:jotpro:v:27:y:2014:i:3:d:10.1007_s10959-012-0452-1
    DOI: 10.1007/s10959-012-0452-1
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    References listed on IDEAS

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    1. Becker-Kern, Peter, 2004. "Random integral representation of operator-semi-self-similar processes with independent increments," Stochastic Processes and their Applications, Elsevier, vol. 109(2), pages 327-344, February.
    2. Makoto Maejima & Ken-iti Sato, 1999. "Semi-Selfsimilar Processes," Journal of Theoretical Probability, Springer, vol. 12(2), pages 347-373, April.
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