# Gamma-type results and other related properties of Poisson processes

@article{Mller1996GammatypeRA, title={Gamma-type results and other related properties of Poisson processes}, author={Jesper M{\o}ller and Sergei A. Zuyev}, journal={Advances in Applied Probability}, year={1996}, volume={28}, pages={662 - 673} }

Families of Poisson processes defined on general state spaces and with the intensity measure scaled by a positive parameter are investigated. In particular, mean value relations with respect to the scale parameter are established and used to derive various Gamma-type results for certain geometric characteristics determined by finite subprocesses. In particular, we deduce Miles' complementary theorem. Applications of the results within stochastic geometry and particularly for random…

## 28 Citations

Decomposition of gamma-distributed domains constructed from Poisson point processes

- Mathematics, Computer ScienceAdvances in Applied Probability
- 2003

In the theoretical development, the classical theorem of Slivnyak is generalized, relating it to the strong Markov property of the Poisson process and to a result of Mecke and Muche (1995).

Variational analysis of functionals of a Poisson process

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- 1998

Let F be a functional of a Poisson point process whose distribution is determined by the intensity measure μ. Considering the expectation EμF as a function on the cone M of positive finite measures…

On the Mixed Poisson Measure

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- 2011

The Papangelou intensity and correlation measure of the mixed Poisson measure are discussed, and by using the Mecke identity, we study a class of birth-death type Dirichlet forms for the mixed…

Stopping sets: Gamma-type results and hitting properties

- MathematicsAdvances in Applied Probability
- 1999

Recently in the paper by Møller and Zuyev (1996), the following Gamma-type result was established. Given n points of a homogeneous Poisson process defining a random figure, its volume is Γ(n,λ)…

Laplace transform identities for the volume of stopping sets based on Poisson point processes

- MathematicsAdvances in Applied Probability
- 2015

We derive Laplace transform identities for the volume content of random stopping sets based on Poisson point processes. Our results are based on anticipating Girsanov identities for Poisson point…

Perturbation analysis of functionals of random measures

- MathematicsAdvances in Applied Probability
- 1995

We use the fact that the Palm measure of a stationary random measure is invariant to phase space change to generalize the light traffic formula initially obtained for stationary processes on a line…

Poisson process Fock space representation, chaos expansion and covariance inequalities

- Mathematics
- 2009

We consider a Poisson process η on an arbitrary measurable space with an arbitrary sigma-finite intensity measure. We establish an explicit Fock space representation of square integrable functions of…

STOPPING SETS: GAMMA-TYPE RESULTS AND Hl'l"l'lNG PROPERTIES

- Mathematics
- 1999

Recently in the paper by M011er and Zuyev (1996), the following Gamma-type result was established. Given n points of a homogeneous Poisson process defining a random figure, its volume is r(n, A)…

Poincaré and weak Poincaré inequalities for the mixed Poisson measure

- Mathematics
- 2014

By using the Mecke identity, we study a class of birth-death type Dirichlet forms associated with the mixed Poisson measure. Both Poincaré and weak Poincaré inequalities are established, while…

Weighted Poisson Cells as Models for Random Convex Polytopes

- Mathematics
- 2013

We introduce a parametric family for random convex polytopes in ℝd which allows for an easy generation of samples for further use, e.g., as random particles in materials modelling and simulation. The…

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