  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.
