Moments and Central Limit Theorems for Some Multivariate Poisson Functionals

Last, Günter; Penrose, Mathew D.; Schulte, Matthias; Thäle, Christoph

doi:10.1017/s0001867800007126

Cited by 44 publications

(91 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The cumulants of I 2 (u) are given by sums over non-flat, connected diagrams (π m , σ ) such that every block of σ has cardinality at least 2; see [51] and [44]. This corresponds to connected multigraphs for which every vertex i ∈…”

Section: Remark 24 (Gaussian Diagrams)mentioning

confidence: 99%

Cluster Expansions for GIBBS Point Processes

Jansen

2019

Adv. Appl. Probab.

View full text Add to dashboard Cite

We provide a sufficient condition for the uniqueness in distribution of Gibbs point processes with non-negative pairwise interaction, together with convergent expansions of the log-Laplace functional, factorial moment densities and factorial cumulant densities (correlation functions and truncated correlation functions). The criterion is a continuum version of a convergence condition by Fernández and Procacci (2007), the proof is based on the Kirkwood-Salsburg integral equations and is close in spirit to the approach by Bissacot, Fernández and Procacci (2010). In addition, we provide formulas for double stochastic integrals with respect to Poisson random measures (not compensated) in terms of multigraphs and pairs of partitions, explaining how to go from cluster expansions to some diagrammatic expansions (Peccati and Taqqu, 2011). We also discuss relations with generating functions for trees, branching processes, Boolean percolation and the random connection model. The presentation is self-contained and requires no preliminary knowledge of cluster expansions.

show abstract

Section: Remark 24 (Gaussian Diagrams)mentioning

confidence: 99%

Cluster Expansions for GIBBS Point Processes

Jansen

2019

Adv. Appl. Probab.

View full text Add to dashboard Cite

show abstract

“…For Poisson hyperplane tessellations many first-and second-order quantities are explicitly available for a broad class of functionals and also a comprehensive central limit theory has been developed over the last 15 years, cf. [21,23,36,58,66] and [64,Chapter 10] as well as the many references cited therein. In the literature, central limit theorems for functionals of Poisson hyperplanes have been considered in two different set-ups.…”

Section: Introductionmentioning

confidence: 99%

“…Alternatively, the intensity is kept fixed, while the size of the observation window is increased. By a simple scaling relation both set-ups are equivalent when homogeneous functionals (such as intrinsic volumes, positive powers of intrinsic volumes or integrals with respect to support measures) of the tessellation are considered, see [36,Corollary 6.2].…”

Section: Introductionmentioning

confidence: 99%

Does a central limit theorem hold for the k-skeleton of Poisson hyperplanes in hyperbolic space?

Herold

Hug

Thäle

2021

Probab. Theory Relat. Fields

Self Cite

View full text Add to dashboard Cite

Poisson processes in the space of $$(d-1)$$ ( d - 1 ) -dimensional totally geodesic subspaces (hyperplanes) in a d-dimensional hyperbolic space of constant curvature $$-1$$ - 1 are studied. The k-dimensional Hausdorff measure of their k-skeleton is considered. Explicit formulas for first- and second-order quantities restricted to bounded observation windows are obtained. The central limit problem for the k-dimensional Hausdorff measure of the k-skeleton is approached in two different set-ups: (i) for a fixed window and growing intensities, and (ii) for fixed intensity and growing spherical windows. While in case (i) the central limit theorem is valid for all $$d\ge 2$$ d ≥ 2 , it is shown that in case (ii) the central limit theorem holds for $$d\in \{2,3\}$$ d ∈ { 2 , 3 } and fails if $$d\ge 4$$ d ≥ 4 and $$k=d-1$$ k = d - 1 or if $$d\ge 7$$ d ≥ 7 and for general k. Also rates of convergence are studied and multivariate central limit theorems are obtained. Moreover, the situation in which the intensity and the spherical window are growing simultaneously is discussed. In the background are the Malliavin–Stein method for normal approximation and the combinatorial moment structure of Poisson U-statistics as well as tools from hyperbolic integral geometry.

show abstract

“…So far the rates of convergence in the Theorem 4 as well as in Lemma 5 are unknown. It seems that the methods applied in the recent papers [36] and [30] can help to answer this open question. Proof of Theorem 2.…”

mentioning

confidence: 99%

Gaussian limits of empirical multiparameter K-functions of homogeneous Poisson processes and tests for complete spatial randomness

Heinrich

2015

Lith Math J

View full text Add to dashboard Cite

Abstract. We prove two functional limit theorems for empirical multiparameter second moment functions (generalizing Ripley's K-function) obtained from a homogeneous Poisson point field observed in an unboundedly expanding convex sampling window W n in R d . The cases of known and unknown (estimated) intensity lead to distinct Gaussian limits and require quite different proofs. Further we determine the limit distributions of the maximal deviation and the integrated squared distance between empirical and true multiparameter second moment function. These results give rise to construct goodness-of-fit tests for checking the hypothesis that a given point pattern is completely spatially random (CSR), i.e. a realization of a homogeneous Poisson process.

show abstract

Moments and Central Limit Theorems for Some Multivariate Poisson Functionals

Cited by 44 publications

References 16 publications

Cluster Expansions for GIBBS Point Processes

Cluster Expansions for GIBBS Point Processes

Does a central limit theorem hold for the k-skeleton of Poisson hyperplanes in hyperbolic space?

Gaussian limits of empirical multiparameter K-functions of homogeneous Poisson processes and tests for complete spatial randomness

Contact Info

Product

Resources

About