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The Dichotomy of Recurrence and Transience of Semi-Lévy Processes

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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.

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Acknowledgements

The authors are grateful to Ken-iti Sato for suggesting to study the dichotomy problem of recurrence and transience of semi-Lévy processes. They also thank two anonymous referees for their detailed comments, by which this paper was improved very much. The helpful comments by Noriyoshi Sakuma are also appreciated.

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Authors and Affiliations

  1. Department of Mathematics, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan

    Makoto Maejima, Taisuke Takamune & Yohei Ueda

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  1. Makoto Maejima
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  2. Taisuke Takamune
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  3. Yohei Ueda
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Correspondence to Makoto Maejima.

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Maejima, M., Takamune, T. & Ueda, Y. The Dichotomy of Recurrence and Transience of Semi-Lévy Processes. J Theor Probab 27, 982–996 (2014). https://doi.org/10.1007/s10959-012-0452-1

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  • DOI: https://doi.org/10.1007/s10959-012-0452-1

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