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Strictly stable distributions on convex cones

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1.	Title	Title of document	Strictly stable distributions on convex cones
2.	Creator	Author's name, affiliation, country	Youri Davydov; University Lille 1
2.	Creator	Author's name, affiliation, country	Ilya Molchanov; University of Bern
2.	Creator	Author's name, affiliation, country	Sergei Zuyev; University of Strathclyde
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	character; convex cone; Laplace transform; LePage series; Lévy measure; point process; Poisson process; random measure; random set; semigroup; stable distribution; union-stability
3.	Subject	Subject classification	60E07; 60B99; 60D05; 60G52; 60G55
4.	Description	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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	British Council - Alliance Française; Swiss National Science Foundation
7.	Date	(YYYY-MM-DD)	2008-02-22
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/487
10.	Identifier	Digital Object Identifier	10.1214/EJP.v13-487
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 13
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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