We present an algorithm for generating Poisson-disk patterns takingO(N) time to generateN points. The method is based on a grid of regions which can contain no more than one point in the final pattern, and uses an explicit model of point arrival times under a uniform Poisson process.

We discuss strong Markov property of Poisson point processes and the related stopping sets. Viewing Poisson process as a set indexed random field, we demonstrate how the martingale technique applies to establish the analogues of the classical results: Doob’s theorem, Wald identity in this multi-dimensional setting. In particular, we show that the famous Slivnyak-Mecke theorem characterising the...

The telegraph process models a random motion with finite velocity and it is usually proposed as an alternative to diffusion models. The process describes the position of a particle moving on the real line, alternatively with constant velocity +v or −v. The changes of direction are governed by an homogeneous Poisson process with rate λ > 0. In this paper, we consider a change point estimation pr...

We consider a Poisson cluster model, motivated by insurance applications. At each claim arrival time, modeled by the point of a homogeneous Poisson process, we start a cluster process which represents the number or amount of payments triggered by the arrival of a claim in a portfolio. The cluster process is a Lévy or truncated compound Poisson process. Given the observations of the process over...

U -statistics of spatial point processes given by a density with respect to a Poisson process are investigated. In the first half of the paper general relations are derived for the moments of the functionals using kernels from the Wiener–Itô chaos expansion. In the second half we obtain more explicit results for a system of U -statistics of some parametric models in stochastic geometry. In the ...

Second order characteristics, in particular Ripley's K-function, are widely used for the statistical analysis of point patterns. We examine a third order analogue, namely the mean number T(R) of R-close triples of points per unit area. Equivalently this is the expected number of R-close point pairs in an R-neighbourhood of a typical point. Various estimators for this function are proposed and c...

Several authors have had apparent success in applying continuous-time point process models to rainfall occurrence sequences. In this paper, it is shown that if rainfall occurrences are interpreted as the events of a point process (and not as a censored sample), the continuous-time point process methodology and estimation procedures are not directly applicable since they fail to account for the ...

We derive sufficient conditions for the mixing of all orders of interacting transformations of a spatial Poisson point process, under a zero-type condition in probability and a generalized adaptedness condition. This extends a classical result in the case of deterministic transformations of Poisson measures. The approach relies on moment and covariance identities for Poisson stochastic integral...

In this paper, we introduce and study a convoluted version of the time fractional Poisson process by taking discrete convolution with respect to space variable in system differential equations that governs its state probabilities. We call introduced as (CFPP). The explicit expression for Laplace transform probabilities are obtained whose inversion yields one-dimensional distribution. Some stati...