Let Kn be the convex hull of i.i.d. random variables distributed according to the standard normal distribution on Rd. We establish variance asymptotics for the re-scaled volume and k-face functional of Kn, k ∈ {0, 1, ..., d − 1}, resolving an open problem. Asymptotic variances and the scaling limit of the boundary of Kn are given in terms of functionals of germ-grain models having parabolic gra...

We study invasion percolation on Aldous’ Poisson-weighted infinite tree, and derive two distinct Markovian representations of the resulting process. One of these is the σ →∞ limit of a representation discovered by Angel et al. [Ann. Appl. Probab. 36 (2008) 420–466]. We also introduce an exploration process of a randomly weighted Poisson incipient infinite cluster. The dynamics of the new proces...

The aim of the paper is to provide an exact approach for generating a Poisson process sampled from a hierarchical CRM, without having to instantiate the infinitely many atoms of the random measures. We use completely random measures (CRM) and hierarchical CRM to define a prior for Poisson processes. We derive the marginal distribution of the resultant point process, when the underlying CRM is m...

Let X be a Poisson point process and K ⊂ R d a measurable set. Construct the Voronoi cells of all points x ∈ X with respect to X, and denote by vX (K) the union of all Voronoi cells with nucleus in K. For K a compact convex set the expectation of the volume difference V (vX (K)) − V (K) and the symmetric difference V (vX (K)△K) is computed. Precise estimates for the variance of both quantities ...

We introduce a probabilistic model for the factorisation of continuous Poisson process rate functions. Our model can be thought of as a topic model for Poisson point processes in which each point is assigned to one of a set of latent rate functions that are shared across multiple outputs. We show that the model brings a means of incorporating structure in point process inference beyond the stat...

We consider optimal stopping of independent sequences. Assuming that the corresponding imbedded planar point processes converge to a Poisson process we introduce some additional conditions which allow to approximate the optimal stopping problem of the discrete time sequence by the optimal stopping of the limiting Poisson process. The optimal stopping of the involved Poisson processes is reduced...

Compound Poisson processes are often useful as approximate models, when describing the occurrence of rare events. In this paper, we develop a method for showing how close such approximations are. Our approach is to use Stein's method directly, rather than by way of declumping and a marked Poisson process; this has conceptual advantages, but entails technical difficulties. Several applications a...