Statistics for Spatial Data.

Stochastic equations in infinite dimensions

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An Introduction To The Theory Of Point Processes

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An introduction to the theory of point processes

The Gibbs point processes (GPP) constitute a large class of point processes with interaction between the points. The interaction can be attractive, repulsive, depending on geometrical features whereas the null interaction is associated with the so-called Poisson point process. In a first part of this mini-course, we present several aspects of finite volume GPP defined on a bounded window in $\mathbb {R}^d$. In a second part, we introduce the more complicated formalism of infinite volume GPP defined on the full space $\mathbb {R}^d$. Existence, uniqueness and non-uniqueness of GPP are non-trivial questions which we treat here with completely self-contained proofs. The DLR equations, the GNZ equations and the variational principle are presented as well. Finally we investigate the estimation of parameters. The main standard estimators (MLE, MPLE, Takacs-Fiksel and variational estimators) are presented and we prove their consistency. For sake of simplicity, during all the mini-course, we consider only the case of finite range interaction and the setting of marked points is not presented.

Introduction to the theory of Gibbs point processes

We establish the existence of stationary Gibbsian point processes for interactions that act on hyperedges between the points. For example, such interactions can depend on Delaunay edges or triangles, cliques of Voronoi cells or clusters of k-nearest neighbors. The classical case of pair interactions is also included. The basic tools are an entropy bound and stationarity.

/pdf/existence-of-gibbsian-point-processes-with-geometry-29gbsn78ct.pdf

Existence of Gibbsian point processes with geometry-dependent interactions

We establish the existence of stationary Gibbsian point processes for interactions that act on hyperedges between the points. For example, such interactions can depend on Delaunay edges or triangles, cliques of Voronoi cells or clusters of $k$-nearest neighbors. The classical case of pair interactions is also included. The basic tools are an entropy bound and stationarity.

We study the (strong-) Gibbsian character on $\mathbb{R}^{\mathbb{Z}^d}$ of the law at time t of an infinite- imensional gradient Brownian diffusion, when the initial distribution is Gibbsian

/pdf/propagation-of-gibbsianness-for-infinite-dimensional-5bv7u4i2g1.pdf

Propagation of Gibbsianness for Infinite-dimensional Gradient Brownian Diffusions

We prove the existence of infinite-volume quermass-interaction processes in a general setting of nonlocally stable interaction and nonbounded convex grains No condition on the parameters of the linear combination of the Minkowski functionals is assumed The only condition is that the square of the random radius of the grain admits exponential moments for all orders Our methods are based on entropy and large deviation tools

/pdf/the-existence-of-quermass-interaction-processes-for-344oj3y1cn.pdf

David Dereudre

Papers

Introduction to the theory of Gibbs point processes

Existence of Gibbsian point processes with geometry-dependent interactions

Existence of Gibbsian point processes with geometry-dependent interactions

Propagation of Gibbsianness for Infinite-dimensional Gradient Brownian Diffusions

The existence of quermass-interaction processes for nonlocally stable interaction and nonbounded convex grains