  Discrete stability extends the classical notion of stability to random
elements in discrete spaces by defining a scaling operation in a randomised
way: an integer is transformed into the corresponding binomial distribution.
Similarly defining the scaling operation as thinning of counting measures we
characterise the corresponding discrete stability property of point processes.
It is shown that these processes are exactly Cox (doubly stochastic Poisson)
processes with strictly stable random intensity measures. We give spectral and
LePage representations for general strictly stable random measures without
assuming their independent scattering. As a consequence, spectral
representations are obtained for the probability generating functional and void
probabilities of discrete stable processes. An alternative cluster
representation for such processes is also derived using the so-called Sibuya
point processes, which constitute a new family of purely random point
processes. The obtained results are then applied to explore stable random
elements in discrete semigroups, where the scaling is defined by means of
thinning of a point process on the basis of the semigroup. Particular examples
include discrete stable vectors that generalise discrete stable random
variables and the family of natural numbers with the multiplication operation,
where the primes form the basis.
