TY - JOUR

T1 - Stability for random measures, point processes and discrete semigroups

AU - Davydov, Youri

AU - Molchanov, Ilya

AU - Zuyev, Sergei

N1 - Copyright: Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2011/8

Y1 - 2011/8

N2 - 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.

AB - 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.

KW - Cluster process

KW - Cox process

KW - Discrete semigroup

KW - Discrete stability

KW - Random measure

KW - Sibuya distribution

KW - Spectral measure

KW - Strict stability

KW - Thinning

UR - http://www.scopus.com/inward/record.url?scp=79960258699&partnerID=8YFLogxK

U2 - 10.3150/10-BEJ301

DO - 10.3150/10-BEJ301

M3 - Article

AN - SCOPUS:79960258699

VL - 17

SP - 1015

EP - 1043

JO - Bernoulli

JF - Bernoulli

SN - 1350-7265

IS - 3

ER -