Completely random signed measures

“…. , ζ d ) and are conditionally independent given ζ, where ζ is a strictly stable non-negative random vector with the Laplace transform (16). Notice that, in general, the components of ξ are dependent DαS random variables unless the spectral measure σ of ζ is supported by the vertices of the simplex ∆ d only.…”

Section: Definition 14mentioning

confidence: 99%

“…Because of this, we first address general strictly stable random measures in Section 2. Note in this relation that, so far, only independently scattered stable measures have received much attention in the literature; see [28] and, more recently, [7,16] on the subject. It should be noted that Cox processes driven by various random measures are often used in spatial statistics (see [17,21,22]).…”

Section: Introductionmentioning

confidence: 99%

Stability for random measures, point processes and discrete semigroups

Davydov¹,

Molchanov²,

Zuyev³

2011

Bernoulli

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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.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ301 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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Statistical models for spatio-temporal extrema and dependencies

Noven¹

2016

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Completely random signed measures

Abstract: Completely random signed measures are defined, characterized and related to Lévy random measures and Lévy bases.

Cited by 3 publications

References 11 publications

Spectral representations of quasi-infinitely divisible processes

Spectral representations of quasi-infinitely divisible processes

Stability for random measures, point processes and discrete semigroups

Statistical models for spatio-temporal extrema and dependencies

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