Strictly stable distributions on convex cones
Davydov, T. and Molchanov, I. and Zuyev, S. (2008) Strictly stable distributions on convex cones. Electronic Journal of Probability, 13. pp. 259321. ISSN 10836489
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Using the LePage representation, a symmetric alphastable 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 unionstable, 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 unionstability 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 unionstable 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 unionstable point process. Since the latter corresponds to a stable law, this yields a limit theorem for normalised sums of random elements with alphastable limit for alpha from (0,1). By using the technique of harmonic analysis on semigroups we characterise distributions of alphastable 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.


Item type: Article ID code: 19772 Dates: DateEvent22 February 2008PublishedKeywords: character, convex cone, Laplace transform, LePage series, Levy measure, point process, Poisson process, random measure, random set, semigroup, stable distribution, unionstability, Probabilities. Mathematical statistics, Statistics and Probability, Statistics, Probability and Uncertainty Subjects: Science > Mathematics > Probabilities. Mathematical statistics Department: Faculty of Science > Mathematics and Statistics > Statistics and Modelling Science Depositing user: Strathprints Administrator Date deposited: 02 Jun 2010 19:10 Last modified: 25 Jun 2021 02:06 URI: https://strathprints.strath.ac.uk/id/eprint/19772