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Stochastic Integral and Series Representations for Strictly Stable Distributions

Author

Listed:
  • Makoto Maejima

    (Keio University)

  • Jan Rosiński

    (University of Tennessee)

  • Yohei Ueda

    (Keio University)

Abstract

In this paper, we find and develop a stochastic integral representation for the class of strictly stable distributions. We establish an explicit relationship between stochastic integral and shot-noise series representations of strictly stable distributions, which shows that the class of distributions representable by stochastic integral is larger than the class representable by a shot-noise series. This inclusion is proper when the stability index $$\alpha $$ α is greater than 1. We also give an explicit description of distributions possessing both representations.

Suggested Citation

  • Makoto Maejima & Jan Rosiński & Yohei Ueda, 2015. "Stochastic Integral and Series Representations for Strictly Stable Distributions," Journal of Theoretical Probability, Springer, vol. 28(3), pages 989-1006, September.
    • Handle: RePEc:spr:jotpro:v:28:y:2015:i:3:d:10.1007_s10959-013-0518-8
    DOI: 10.1007/s10959-013-0518-8
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    References listed on IDEAS

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    1. Ichifuji, Ken & Maejima, Makoto & Ueda, Yohei, 2010. "Fixed points of mappings of infinitely divisible distributions on," Statistics & Probability Letters, Elsevier, vol. 80(17-18), pages 1320-1328, September.
    2. Sato, Ken-iti & Yamazato, Makoto, 1984. "Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type," Stochastic Processes and their Applications, Elsevier, vol. 17(1), pages 73-100, May.
    3. Ken-iti Sato & Yohei Ueda, 2013. "Weak Drifts of Infinitely Divisible Distributions and Their Applications," Journal of Theoretical Probability, Springer, vol. 26(3), pages 885-898, September.
    4. Ken-iti Sato, 2013. "Inversions of Infinitely Divisible Distributions and Conjugates of Stochastic Integral Mappings," Journal of Theoretical Probability, Springer, vol. 26(4), pages 901-931, December.
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