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Inversions of Infinitely Divisible Distributions and Conjugates of Stochastic Integral Mappings

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  • Ken-iti Sato

    (Hachiman-yama 1101-5-103)

Abstract

The dual of an infinitely divisible distribution on ℝ d without Gaussian part defined in Sato (ALEA Lat. Am. J. Probab. Math. Statist. 3:67–110, 2007) is renamed to the inversion. Properties and characterization of the inversion are given. A stochastic integral mapping is a mapping μ=Φ f ρ of ρ to μ in the class of infinitely divisible distributions on ℝ d , where μ is the distribution of an improper stochastic integral of a nonrandom function f with respect to a Lévy process on ℝ d with distribution ρ at time 1. The concept of the conjugate is introduced for a class of stochastic integral mappings and its close connection with the inversion is shown. The domains and ranges of the conjugates of three two-parameter families of stochastic integral mappings are described. Applications to the study of the limits of the ranges of iterations of stochastic integral mappings are made.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:26:y:2013:i:4:d:10.1007_s10959-012-0420-9
    DOI: 10.1007/s10959-012-0420-9
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    References listed on IDEAS

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    1. Takahiro Aoyama & Makoto Maejima & Jan Rosiński, 2008. "A Subclass of Type G Selfdecomposable Distributions on ℝ d," Journal of Theoretical Probability, Springer, vol. 21(1), pages 14-34, March.
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    Cited by:

    1. 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.

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