Moment formulae for general point processes

Decreusefond, Laurent; Flint, Ian

doi:10.1016/j.crma.2013.11.016

Cited by 12 publications

(23 citation statements)

References 16 publications

(19 reference statements)

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“…UnderP N , the random variables ω and ζ are independent. Equation (14) means that the marginal distribution of ζ tends to M (assumed to be a probability measure at the very beginning of this construction). Moreover, we already know that P N converges in distribution to P. Hence,P N tends to P ⊗ M as N goes to infinity.…”

Section: Proofs Of Sectionmentioning

confidence: 99%

Malliavin and Dirichlet structures for independent random variables

Decreusefond

Halconruy

2019

Stochastic Processes and their Applications

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On any denumerable product of probability spaces, we construct a Malliavin gradient and then a divergence and a number operator. This yields a Dirichlet structure which can be shown to approach the usual structures for Poisson and Brownian processes. We obtain versions of almost all the classical functional inequalities in discrete settings which show that the Efron-Stein inequality can be interpreted as a Poincaré inequality or that the Hoeffding decomposition of U -statistics can be interpreted as an avatar of the Clark representation formula. Thanks to our framework, we obtain a bound for the distance between the distribution of any functional of independent variables and the Gaussian and Gamma distributions.

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Section: Proofs Of Sectionmentioning

confidence: 99%

Malliavin and Dirichlet structures for independent random variables

Decreusefond

Halconruy

2019

Stochastic Processes and their Applications

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“…The identity (2.2) has been rewritten in the langage of sums over partitions, and extended to Poisson stochastic integrals of random integrands in Proposition 3.1 of [18], and further extended to point processes admitting a Panpangelou intensity in Theorem 3.1 of [5], see also [4]. In the sequel, given z n = (z 1 , .…”

Section: Moment Identitiesmentioning

confidence: 99%

Moments of k-hop counts in the random-connection model

Privault

2019

J. Appl. Probab.

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We derive moment identities for the stochastic integrals of multiparameter processes in a random-connection model based on a point process admitting a Papangelou intensity. Those identities are written using sums over partitions, and they reduce to sums over non-flat partition diagrams in case the multiparameter processes vanish on diagonals. As an application, we obtain general identities for the moments of k-hop counts in the random-connection model, which simplify the derivations available in the literature.

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“…Similar formulas to (10), but under a stronger assumption of a product form of the driving function f of F, were derived in [3] using the Georgii-Nguyen-Zessin formula [1]. As an application consider processes µ a with densities p a w.r.t.…”

Section: U-statisticsmentioning

confidence: 99%

“…Using a special class of functionals called U -statistics closed formulas for mixed moments of functionals are obtained. Similar formulas, but under a stronger assumption of a product form of the driving function of the functional, were derived in [3] using the Georgii-Nguyen-Zessin formula. In processes with densities the key characteristics are the correlation functions [4] of arbitrary order which are dual to kernel functions of the density as a function of the Poisson process.…”

Section: Introductionmentioning

confidence: 99%

Interaction Processes for Unions of Facets, the Asymptotic Behaviour with Increasing Intensity

Vecera

Beneš

2016

Methodol Comput Appl Probab

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In the series of models with interacting particles in stochastic geometry, a further contribution presents the facet process which is defined in arbitrary Euclidean dimension. In 2D, 3D specially it is a process of interacting segments, flat surfaces, respectively. Its investigation is based on the theory of functionals of finite spatial point processes given by a density with respect to a Poisson process. The methodology based on L 2 expansion of the covariance of functionals of Poisson process is developed for U -statistics of facet intersections which are building blocks of the model. The importance of the concept of correlation functions of arbitrary order is emphasized. Some basic properties of facet processes, such as local stability and repulsivness are shown and a standard simulation algorithm mentioned. Further the situation when the intensity of the process tends to infinity is studied. In the case of Poisson processes a central limit theorem follows from recent results of Wiener-Ito theory. In the case of non-Poisson processes we restrict to models with finitely many orientations. Detailed analysis of correlation functions exhibits various asymptotics for different combination of U -statistics and submodels of the facet process.

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Moment formulae for general point processes

Cited by 12 publications

References 16 publications

Malliavin and Dirichlet structures for independent random variables

Malliavin and Dirichlet structures for independent random variables

Moments of k-hop counts in the random-connection model

Interaction Processes for Unions of Facets, the Asymptotic Behaviour with Increasing Intensity

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