For d ≥ 3, we construct a non-randomized, fair and translationequivariant allocation of Lebesgue measure to the points of a standard Poisson point process in Rd, defined by allocating to each of the Poisson points its basin of attraction with respect to the flow induced by a gravitational force field exerted by the points of the Poisson process. We prove that this allocation rule is economical ...

Tra c using the Fractal-Shot-Noise-Driven Poisson Process Bong K. Ryu and Steven B. Lowen Department of Electrical Engineering and Center for Telecommunications Research Columbia University, New York, NY 10027, U.S.A. fryu, [email protected] Abstract Presented in this paper is the fractal-shot-noisedriven Poisson process (FSNDP), a self-similar process which permits parsimonious modeling...

We de ne a family of random trees in the plane. Their nodes of level k; k = 0; : : : ;m are the points of a homogeneous Poisson point process k, whereas their arcs connect nodes of level k and k + 1, according to the least distance principle: if V denotes the Voronoi cell w.r.t. k+1 with nucleus x, where x is a point of k+1, then there is an arc connecting x to all the points of k which belong ...

The lilypond model on a point process in d-space is a growth-maximal system of nonoverlapping balls centred at the points. We establish central limit theorems for the total volume and the number of components of the lilypond model on a sequence of Poisson or binomial point processes on expanding windows. For the lilypond model over a homogeneous Poisson process, we give subexponentially decayin...

Motivated by sampling problems in forestry and related fields, we suggest a spatial sampling scheme for estimating the intensity of a point process. The technique is related to the ‘wandering quarter’ method. In applications where the cost of identifying random points is high relative to the cost of taking measurements, for example when identification involves travelling within a large region, ...

Continuum percolation models in which pairs of points of a two-dimensional Poisson point process are connected if they are within some range of each other have been extensively studied. This paper considers a variation in which a connection between two points depends not only on their Euclidean distance, but also on the positions of all other points of the point process. This model has been rec...

We consider a process that starts at height y, stays there for a time X0 ∼ exp(y) when it drops to a level Z1 ∼ U(0, y). Thereafter it stays at level Zn for time Xn ∼ exp(Zn), then drops to a level Zn+1 ∼ U(0, Zn). We investigate properties of this process, as well as the Poisson hyperbolic process which is obtained by randomizing the starting point y of the above process. This process is assoc...