Strictly stable distributions on convex cones

T. Davydov, I. Molchanov, S. Zuyev

    Research output: Contribution to journalArticle

    37 Citations (Scopus)

    Abstract

    Using the LePage representation, a symmetric alpha-stable random element in Banach space B with alpha from (0,2) can be represented as a sum of points of a Poisson process in B. This point process is union-stable, i.e. the union of its two independent copies coincides in distribution with the rescaled original point process. This shows that the classical definition of stable random elements is closely related to the union-stability property of point processes. These concepts make sense in any convex cone, i.e. in a semigroup equipped with multiplication by numbers, and lead to a construction of stable laws in general cones by means of the LePage series. We prove that random samples (or binomial point processes) in rather general cones converge in distribution in the vague topology to the union-stable Poisson point process. This convergence holds also in a stronger topology, which implies that the sums of points converge in distribution to the sum of points of the union-stable point process. Since the latter corresponds to a stable law, this yields a limit theorem for normalised sums of random elements with alpha-stable limit for alpha from (0,1). By using the technique of harmonic analysis on semigroups we characterise distributions of alpha-stable random elements and show how possible values of the characteristic exponent alpha relate to the properties of the semigroup and the corresponding scaling operation, in particular, their distributivity properties. It is shown that several conditions imply that a stable random element admits the LePage representation. The approach developed in the paper not only makes it possible to handle stable distributions in rather general cones (like spaces of sets or measures), but also provides an alternative way to prove classical limit theorems and deduce the LePage representation for strictly stable random vectors in Banach spaces.
    LanguageEnglish
    Pages259-321
    Number of pages62
    JournalElectronic Journal of Probability
    Volume13
    Publication statusPublished - 22 Feb 2008

    Fingerprint

    Stable Distribution
    Convex Cone
    Strictly
    Random Element
    Point Process
    Union
    Stable Laws
    Cone
    Semigroup
    Limit Theorems
    Banach space
    Topology
    Converge
    Imply
    Distributivity
    Characteristic Exponents
    Poisson Point Process
    Convex cone
    Stable distribution
    Classical Limit

    Keywords

    • character
    • convex cone
    • Laplace transform
    • LePage series
    • Levy measure
    • point process
    • Poisson process
    • random measure
    • random set
    • semigroup
    • stable distribution
    • union-stability

    Cite this

    Davydov, T., Molchanov, I., & Zuyev, S. (2008). Strictly stable distributions on convex cones. Electronic Journal of Probability, 13, 259-321.
    Davydov, T. ; Molchanov, I. ; Zuyev, S. / Strictly stable distributions on convex cones. In: Electronic Journal of Probability. 2008 ; Vol. 13. pp. 259-321.
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    Davydov, T, Molchanov, I & Zuyev, S 2008, 'Strictly stable distributions on convex cones' Electronic Journal of Probability, vol. 13, pp. 259-321.

    Strictly stable distributions on convex cones. / Davydov, T.; Molchanov, I.; Zuyev, S.

    In: Electronic Journal of Probability, Vol. 13, 22.02.2008, p. 259-321.

    Research output: Contribution to journalArticle

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    T1 - Strictly stable distributions on convex cones

    AU - Davydov, T.

    AU - Molchanov, I.

    AU - Zuyev, S.

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    N2 - Using the LePage representation, a symmetric alpha-stable random element in Banach space B with alpha from (0,2) can be represented as a sum of points of a Poisson process in B. This point process is union-stable, i.e. the union of its two independent copies coincides in distribution with the rescaled original point process. This shows that the classical definition of stable random elements is closely related to the union-stability property of point processes. These concepts make sense in any convex cone, i.e. in a semigroup equipped with multiplication by numbers, and lead to a construction of stable laws in general cones by means of the LePage series. We prove that random samples (or binomial point processes) in rather general cones converge in distribution in the vague topology to the union-stable Poisson point process. This convergence holds also in a stronger topology, which implies that the sums of points converge in distribution to the sum of points of the union-stable point process. Since the latter corresponds to a stable law, this yields a limit theorem for normalised sums of random elements with alpha-stable limit for alpha from (0,1). By using the technique of harmonic analysis on semigroups we characterise distributions of alpha-stable random elements and show how possible values of the characteristic exponent alpha relate to the properties of the semigroup and the corresponding scaling operation, in particular, their distributivity properties. It is shown that several conditions imply that a stable random element admits the LePage representation. The approach developed in the paper not only makes it possible to handle stable distributions in rather general cones (like spaces of sets or measures), but also provides an alternative way to prove classical limit theorems and deduce the LePage representation for strictly stable random vectors in Banach spaces.

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    KW - semigroup

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    ER -

    Davydov T, Molchanov I, Zuyev S. Strictly stable distributions on convex cones. Electronic Journal of Probability. 2008 Feb 22;13:259-321.