Project a collection of points on the high-dimensional sphere onto a random direction. If most of the points are sufficiently far from one another in an appropriate sense, the projection is locally close in distribution to the Poisson point process.

We construct a non-decreasing pure jump Markov process, whose jump measure heavily depends on the values taken by the process. We determine the singularity spectrum of this process, which turns out to be random and to depend locally on the values taken by the process. The result relies on fine properties of the distribution of Poisson point processes and on ubiquity theorems.

Continuum percolation models in which each point of a two-dimensional Poisson point process is the centre of a disc of given (or random) radius r , have been extensively studied. In this paper, we consider the generalization in which a deterministic algorithm (given the points of the point process) places the discs on the plane, in such a way that each disc covers at least one point of the poin...

In many operations research applications we want to know if a stochastic point process (series of events) can be modeled as a renewal process or a Poisson process. In other words, we want to know if the intervals between points are approximately lid (independent and identically distributed) and, if so, whether they are approximately exponentially distributed. For example, we may be considering ...

Motivated by the problem of minefield detection, we investigate the problem of classifying mixtures of spatial point processes. In particular we are interested in testing the hypothesis that a given dataset was generated by a Poisson process versus a mixture of a Poisson process and a hard-core Strauss process. We propose testing this hypothesis by comparing the evidence for each model by using...

In analyzing genetic linkage data it is common to assume that the locations of crossovers along a chromosome follow a Poisson process, whereas it has long been known that this assumption does not fit the data. In many organisms it appears that the presence of a crossover inhibits the formation of another nearby, a phenomenon known as "interference." We discuss several point process models for r...

In constructing a time histogram of the event sequences derived from a nonstationary point process, we wish to determine the bin width such that the mean squared error of the histogram from the underlying rate of occurrence is minimized. We find that the optimal bin widths obtained for a doubly stochastic Poisson process and a sinusoidally regulated Poisson process exhibit different scaling rel...

In the present paper we consider the mathematical model of secondary processes and give a general and rigorous method for solving special problems. We do not suppose that the basic “process” is a time-process but consider the problem more generally, i.e. we replace the time axis by an abstract space where the random points are distributed. The idea of our general method was suggested by a lectu...

In many contexts, such as queueing theory, spatial statistics, geostatistics and meteorology, data are observed at irregular spatial positions. One model of this situation is to consider the observation points as generated by a Poisson Process. Under this assumption, we study the limit behavior of the partial sums of the Marked Point Process {(ti, X(ti))}, where X(t) is a stationary random fiel...