scholarly journals Cluster Expansions for GIBBS Point Processes

2019 ◽  
Vol 51 (4) ◽  
pp. 1129-1178 ◽  
Author(s):  
S. Jansen
Keyword(s):  
Correlation Functions ◽  
Point Processes ◽  
Branching Processes ◽  
Factorial Moment ◽  
Convergence Condition ◽  
Random Measures ◽  
Cluster Expansions ◽  
Gibbs Point Processes ◽  
Random Connection Model ◽  
Boolean Percolation

AbstractWe 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 cumulants of 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.

scholarly journals Existence of Gibbs point processes with stable infinite range interaction

2020 ◽  
Vol 57 (3) ◽  
pp. 775-791
Author(s):  
David Dereudre ◽  
Thibaut Vasseur
Keyword(s):  
Point Processes ◽  
Existence Results ◽  
Pairwise Interaction ◽  
Range Interaction ◽  
Pairwise Interactions ◽  
Gibbs Point Processes ◽  
Infinite Range Interactions ◽  
Finite Configuration ◽  
Multi Body ◽  
Infinite Range

AbstractWe provide a new proof of the existence of Gibbs point processes with infinite range interactions, based on the compactness of entropy levels. Our main existence theorem holds under two assumptions. The first one is the standard stability assumption, which means that the energy of any finite configuration is superlinear with respect to the number of points. The second assumption is the so-called intensity regularity, which controls the long range of the interaction via the intensity of the process. This assumption is new and introduced here since it is well adapted to the entropy approach. As a corollary of our main result we improve the existence results by Ruelle (1970) for pairwise interactions by relaxing the superstabilty assumption. Note that our setting is not reduced to pairwise interaction and can contain infinite-range multi-body counterparts.

scholarly journals Fast approximation of the intensity of Gibbs point processes

2012 ◽  
Vol 6 (0) ◽  
pp. 1155-1169 ◽  
Author(s):  
Adrian Baddeley ◽  
Gopalan Nair
Keyword(s):  
Point Processes ◽  
Gibbs Point Processes

Parametric inference in Markov branching processes with time-dependent random immigration rate

1985 ◽  
Vol 22 (03) ◽  
pp. 503-517
Author(s):  
Helmut Pruscha
Keyword(s):  
Point Processes ◽  
Traditional Approach ◽  
Branching Processes ◽  
External Source ◽  
Time Dependent ◽  
Parametric Inference ◽  
Asymptotic Results ◽  
Subcritical Case ◽  
Immigration Rate ◽  
Pearson Type

The present paper deals with continuous-time Markov branching processes allowing immigration. The immigration rate is allowed to be random and time-dependent where randomness may stem from an external source or from state-dependence. Unlike the traditional approach, we base the analysis of these processes on the theory of multivariate point processes. Using the tools of this theory, asymptotic results on parametric inference are derived for the subcritical case. In particular, the limit distributions of some parametric estimators and of Pearson-type statistics for testing simple and composite hypotheses are established.

Rarefactions of compound point processes

1984 ◽  
Vol 21 (04) ◽  
pp. 710-719
Author(s):  
Richard F. Serfozo
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Classical Result ◽  
Point Processes ◽  
Rare Events ◽  
Random Measure ◽  
Bernoulli Trials ◽  
Random Measures ◽  
Infinitely Divisible ◽  
Independent Increments

The Poisson process is regarded as a point process of rare events because of the classical result that the number of successes in a sequence of Bernoulli trials is asymptotically Poisson as the probability of a success tends to 0. It is shown that this rareness property of the Poisson process is characteristic of any infinitely divisible point process or random measure with independent increments. These processes and measures arise as limits of certain rarefactions of compound point processes: purely atomic random measures with uniformly null atom sizes. Examples include thinnings and partitions of point processes.

Limit theorems for point processes generated in a general branching process

1981 ◽  
Vol 13 (04) ◽  
pp. 650-668 ◽  
Author(s):  
Martin Härnqvist
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Branching Processes ◽  
Special Problem ◽  
Convergence Theory ◽  
Regularity Conditions ◽  
Convergence Results ◽  
Mixed Poisson Process ◽  
Proper Scaling ◽  
Extra Point

With the general convergence theory for branching processes as basis a special problem is studied. An extra point process of events during life is assigned to each realised individual, and the behaviour of the superposition of such point processes in action is studied as the population grows. With the proper scaling and under some regularity conditions the superposition is shown to converge in distribution to a Poisson process. Another scaling gives rise to a mixed Poisson process as limit. Established weak convergence techniques for point processes are applied, together with some recent strong convergence results for branching processes.

Modelling Tree Roots in Mixed Forest Stands by Inhomogeneous Marked Gibbs Point Processes

2009 ◽  
Vol 51 (3) ◽  
pp. 522-539 ◽  
Author(s):  
Stefanie Eckel ◽  
Frank Fleischer ◽  
Pavel Grabarnik ◽  
Marian Kazda ◽  
Aila Särkkä ◽  
...  
Keyword(s):  
Point Processes ◽  
Mixed Forest ◽  
Forest Stands ◽  
Tree Roots ◽  
Gibbs Point Processes
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