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**Harvard**

Zuyev, S., Davydov, Y. och Molchanov, I. (2011) *Stability for random measures, point processes and discrete semigroups.*.

** BibTeX **

@article{

Zuyev2011,

author={Zuyev, Sergei and Davydov, Y. and Molchanov, I.},

title={Stability for random measures, point processes and discrete semigroups.},

journal={BERNOULLI},

issn={1350-7265},

volume={17},

issue={3},

pages={1015-1043},

abstract={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. },

year={2011},

keywords={cluster process, Cox process, discrete semigroup, discrete stability, random measure, Sibuya distribution, spectral measure, strict stability, thinning},

}

** RefWorks **

RT Journal Article

SR Electronic

ID 144570

A1 Zuyev, Sergei

A1 Davydov, Y.

A1 Molchanov, I.

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

YR 2011

JF BERNOULLI

SN 1350-7265

VO 17

IS 3

SP 1015

OP 1043

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.

LA eng

LK http://dx.doi.org/10.3150/10-BEJ301

OL 30