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Fractional Integrals and Extensions of Selfdecomposability

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Lévy Matters I

Part of the book series: Lecture Notes in Mathematics ((LEVY,volume 2001))

Abstract

After characterizations of the class L of selfdecomposable distributions on \({\mathbb{R}}^{d}\) are recalled, the classes K p, α and L p, α with two continuous parameters 0 < p < ∞ and − ∞ < α < 2 satisfying \({K}_{1,0} = {L}_{1,0} = L\) are introduced as extensions of the class L. They are defined as the classes of distributions of improper stochastic integrals ∫0 ∞ − f(s)dX s (ρ), where f(s) is an appropriate non-random function and X s (ρ) is a Lévy process on \({\mathbb{R}}^{d}\) with distribution ρ at time 1. The description of the classes is given by characterization of their Lévy measures, using the notion of monotonicity of order p based on fractional integrals of measures, and in some cases by addition of the condition of zero mean or some weaker conditions that are newly introduced – having weak mean 0 or having weak mean 0 absolutely. The class L n, 0 for a positive integer n is the class of n times selfdecomposable distributions. Relations among the classes are studied. The limiting classes as p → ∞ are analyzed. The Thorin class T, the Goldie–Steutel–Bondesson class B, and the class L ∞ of completely selfdecomposable distributions, which is the closure (with respect to convolution and weak convergence) of the class \(\mathfrak{S}\) of all stable distributions, appear in this context. Some subclasses of the class L ∞ also appear. The theory of fractional integrals of measures is built. Many open questions are mentioned.

AMS Subject Classification 2000

Primary: 60E07, 60H05

Secondary: 26A33, 60G51, 62E10, 62H05

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Authors and Affiliations

  1. Hachiman-yama 1101-5-103, Tenpaku-Ku, Nagoya, 468-0074, Japan

    Ken-iti Sato

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  1. Ken-iti Sato
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Correspondence to Ken-iti Sato .

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Editors and Affiliations

  1. Thiele Centre, Department of Mathematics, Aarhus University, Ny Munkegade 118, Aarhus C, 8000, Denmark

    Ole E Barndorff-Nielsen

  2. , Lab de Probabilités & Modèles Aléatoires, Univ Paris 6 - Pierre et Marie Curie, Place Jussieu 4, Paris Cedex 05, 75252, France

    Jean Bertoin

  3. Pierre et Marie Curie, Institut de Mathématiques de Jussieu, Université Paris VI -, Place Jussieu 4, Paris, Cedex 05, 75252, France

    Jean Jacod

  4. Center for Mathematical Sciences, Munich University of Technology, Boltzmannstraße 3, Garching, 85748, Germany

    Claudia Klüppelberg

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Sato, Ki. (2010). Fractional Integrals and Extensions of Selfdecomposability. In: Barndorff-Nielsen, O., Bertoin, J., Jacod, J., Klüppelberg, C. (eds) Lévy Matters I. Lecture Notes in Mathematics(), vol 2001. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14007-5_1

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