Strictly stable distributions on convex cones
Youri Davydov, University Lille 1
Ilya Molchanov, University of Bern
Sergei Zuyev, University of Strathclyde
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.
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