Stability for random measures, point processes and discrete semigroups

A scaling operation on non-negative integers can be defined in a randomised way by transforming an integer into the corresponding binomial distribution with success probability being the scaling factor. We explore a similar (thinning) operation defined on counting measures and characterise the corresponding discrete stablility property of point processes. It is shown that these processes are exactly Cox (doubly stochastic Poisson) processes with strictly stable random intensity measures. The paper contains spectral and LePage representations for strictly stable measures and characterises some special cases, e.g. independently scattered measures. As consequence, spectral representations are provided for the probability generating functional and void probabilities of discrete stable processes. An alternative cluster representation for discrete stable processes is also derived using the so-called Sibuya point processes that 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 the one-dimensional case of discrete random variables studied by Steutel and van Harn (1979) and the family of natural numbers with the multiplication operation, where the primes form the basis.