Sergei Zuyev, 
Mathematical Sciences, 
Chalmers University of Technology

Stability for random measures, point processes and discrete semigroups

Abstract

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 stablility 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. The obtained results are then applied to explore stable 
random elements in discrete semigroups admitting a countable basis, in particular,
 the family of natural numbers with the multiplication operation, where the basis 
is the set of primes.