Stable random measures and stable point processes

Stable distributions play fundamental role in probability since they necessarily arise in variety of limit theorems. The definition is based on two operations: addition and scaling by positive numbers. However, in discrete semigroups scaling by arbitrary positive factor cannot be defined and is replaced by a stochastic operation which gives rise to the corresponding stable random elements. Particular examples include discrete stable non-negative random integers studied by Steutel and van Harn (1979). A scaling operation here can be defined 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 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 obtain spectral and LePage representations for strictly stable measures and characterises some special cases, e.g. independently scattered measures. 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. Statistical inference for stable processes is also discussed.