Strictly stable distributions on convex cones

TL;DR

This paper extends the theory of strictly stable distributions to convex cones using the LePage representation, characterizing their properties and limit theorems in a general algebraic setting.

Contribution

It introduces a novel framework for stable laws in convex cones via LePage series and harmonic analysis, broadening the scope beyond Banach spaces.

Findings

Stable laws in convex cones can be constructed using LePage series.

Limit theorems show convergence of random samples to union-stable Poisson processes.

Characterization of $lpha$-stable distributions based on semigroup properties.

Abstract

Using the LePage representation, a strictly stable random element in a Banach space with $\alpha\in(0,2)$ can be represented as a sum of points of a Poisson process. This point process is union-stable, i.e. the union of its two independent copies coincides in distribution with the rescaled original point process. These concepts makes sense in any convex cone, i.e. in a commutative semigroup equipped with multiplication by numbers, and lead to a construction of stable laws in general cones by means of the LePage series. The corresponding limit theorem shows that random samples (or binomial point processes) converge in distribution to the union-stable Poisson point process, and so yields a limit theorem for normalised sums of random elements with $\alpha$-stable limit for $\alpha\in(0,1)$. By using the technique of harmonic analysis on semigroups we characterise distributions of $\alpha$-stable random elements and show how possible values of $\alpha$ relate to the properties of the semigroup and the corresponding scaling operation, in particular, their distributivity properties. 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.