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Semi-Selfsimilar Processes

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  • Makoto Maejima

    (Keio University)

  • Ken-iti Sato

Abstract

A notion of semi-selfsimilarity of R d -valued stochastic processes is introduced as a natural extension of the selfsimilarity. Several topics on semi-selfsimilar processes are studied: the existence of the exponent for semi-selfsimilar processes; characterization of semi-selfsimilar processes as scaling limits; relationship between semi-selfsimilar processes with independent increments and semi-selfdecomposable distributions, and examples; construction of semi-selfsimilar processes with stationary increments; and extension of the Lamperti transformation. Semi-stable processes where all joint distributions are multivariate semi-stable are also discussed in connection with semi-selfsimilar processes. A wide-sense semi-selfsimilarity is defined and shown to be reducible to semi-selfsimilarity.

Suggested Citation

  • Makoto Maejima & Ken-iti Sato, 1999. "Semi-Selfsimilar Processes," Journal of Theoretical Probability, Springer, vol. 12(2), pages 347-373, April.
    • Handle: RePEc:spr:jotpro:v:12:y:1999:i:2:d:10.1023_a:1021621926463
    DOI: 10.1023/A:1021621926463
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    References listed on IDEAS

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    1. Krzysztof Burnecki & Makoto Maejima & Aleksander Weron, 1997. "The Lamperti transformation for self-similar processes," HSC Research Reports HSC/97/02, Hugo Steinhaus Center, Wroclaw University of Technology.
    2. Rajput, Balram S. & Rama-Murthy, Kavi, 1987. "Spectral representation of semistable processes, and semistable laws on Banach spaces," Journal of Multivariate Analysis, Elsevier, vol. 21(1), pages 139-157, February.
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    Citations

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    Cited by:

    1. 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.
    2. Peter Kern & Mark M. Meerschaert & Yimin Xiao, 2018. "Asymptotic Behavior of Semistable Lévy Exponents and Applications to Fractal Path Properties," Journal of Theoretical Probability, Springer, vol. 31(1), pages 598-617, March.
    3. Kenneth J. Falconer, 2002. "Tangent Fields and the Local Structure of Random Fields," Journal of Theoretical Probability, Springer, vol. 15(3), pages 731-750, July.
    4. Tomasz Luks & Yimin Xiao, 2020. "Multiple Points of Operator Semistable Lévy Processes," Journal of Theoretical Probability, Springer, vol. 33(1), pages 153-179, March.
    5. Takahashi, Hiroshi & Tamura, Yozo, 2023. "Diffusion processes in Brownian environments on disconnected selfsimilar fractal sets in R," Statistics & Probability Letters, Elsevier, vol. 193(C).
    6. Toshiro Watanabe, 2002. "Shift Self-Similar Additive Random Sequences Associated with Supercritical Branching Processes," Journal of Theoretical Probability, Springer, vol. 15(3), pages 631-665, July.
    7. Toshiro Watanabe, 2016. "Escape Rates for Multidimensional Shift Self-similar Additive Sequences," Journal of Theoretical Probability, Springer, vol. 29(3), pages 896-921, September.

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