Electronic Journal of Probability

Strictly stable distributions on convex cones

Youri Davydov, Ilya Molchanov, and Sergei Zuyev

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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.

Article information

Source
Electron. J. Probab., Volume 13 (2008), paper no. 11, 259-321.

Dates
Accepted: 22 February 2008
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819084

Digital Object Identifier
doi:10.1214/EJP.v13-487

Mathematical Reviews number (MathSciNet)
MR2386734

Zentralblatt MATH identifier
1196.60028

Subjects
Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60B99: None of the above, but in this section 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G52: Stable processes 60G55: Point processes

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

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Davydov, Youri; Molchanov, Ilya; Zuyev, Sergei. Strictly stable distributions on convex cones. Electron. J. Probab. 13 (2008), paper no. 11, 259--321. doi:10.1214/EJP.v13-487. https://projecteuclid.org/euclid.ejp/1464819084


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