Strictly stable distributions on convex cones

Abstract

Using the LePage representation, a symmetric α-stable random element in Banach space B with α ∈ (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 makes 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 α-stable limit for α ∈ (0, 1).By using the technique of harmonic analysis on semigroups we characterise distributions of α-stable random elements and show how possible values of the characteristic exponent α 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.

Freezing and Decorated Poisson Point Processes

“…Brunet and Derrida [16] (p. 18) conjectured that this property is equivalent to (SDP) with Z = 0. This can be proved using a representation of the Laplace functional for infinitely divisible processes (see Maillard [44]; implicitely, this also appears in [21]). It is fairly simple to see that exponentialc-stability is equivalent to the uniqueness of the support of L ξ [ f | x] up to translation with g (x) = Gum (cx) (cf.…”

Convergence to Stable Laws in the Space D

“…Very general results in the case α ∈ (0, 1) can be found in [6]. sequence of finite dimensional vectors whose common distribution is multivariate regularly varying, then the sum n i=1 X i , suitably centered and normalized, converges to an α-stable distribution.…”

Extremes of independent stochastic processes: a point process approach

“…As noticed by Davis and Mikosch [4], the proof remains valid for a complete separable metric space E if we change vague convergence by -weak convergence. See also Theorem 4.3 in Davydov, Molchanov and Zuyev [5].…”