Random sets with long-range dependence can be generated using a Boolean model with power-law grain sizes. We study thinnings of such Boolean models which have the hard-core property that no grains overlap in the resulting germ‒grain model. A fundamental question is whether long-range dependence is preserved under such thinnings. To answer this question, we study four natural thinnings of a Poisson germ‒grain model where the grains are spheres with a regularly varying size distribution. We show that a thinning which favors large grains preserves the slow correlation decay of the original model, whereas a thinning which favors small grains does not. Our most interesting finding concerns the case where only disjoint grains are retained, which corresponds to the well-known Matérn type-I thinning. In the resulting germ‒grain model, typical grains have exponentially small sizes, but rather surprisingly, the long-range dependence property is still present. As a byproduct, we obtain new mechanisms for generating homogeneous and isotropic random point configurations having a power-law correlation decay.

We study semi-infinite paths of the radial spanning tree (RST) of a Poisson point process in the plane. We first show that the expectation of the number of intersection points between semi-infinite paths and the sphere with radius r grows sublinearly with r. Then we prove that in each (deterministic) direction there exists, with probability 1, a unique semi-infinite path, framed by an infinite number of other semi-infinite paths of close asymptotic directions. The set of (random) directions in which there is more than one semi-infinite path is dense in [0, 2π). It corresponds to possible asymptotic directions of competition interfaces. We show that the RST can be decomposed into at most five infinite subtrees directly connected to the root. The interfaces separating these subtrees are studied and simulations are provided.

This paper presents a powerful characterisation for the structure of internal vertices of the STIT's I-segments. The characterisation allows certain mathematical analyses to be performed easily. We demonstrate this by deriving new results for various topological properties of the tessellation: for example, the numbers of various types of edge and cell side within the typical I-segment. The characterisation also provides a tool for the calculations of metric properties of the tessellation; many new length distributions and frame-coverage results are given.

We investigate the ray-length distributions for two different rectangular versions of Gilbert's tessellation (see Gilbert (1967)). In the full rectangular version, lines extend either horizontally (east- and west-growing rays) or vertically (north- and south-growing rays) from seed points which form a Poisson point process, each ray stopping when another ray is met. In the half rectangular version, east- and south-growing rays do not interact with west and north rays. For the half rectangular tessellation, we compute analytically, via recursion, a series expansion for the ray-length distribution, whilst, for the full rectangular version, we develop an accurate simulation technique, based in part on the stopping-set theory for Poisson processes (see Zuyev (1999)), to accomplish the same. We demonstrate the remarkable fact that plots of the two distributions appear to be identical when the intensity of seeds in the half model is twice that in the full model. In this paper we explore this coincidence, mindful of the fact that, for one model, our results are from a simulation (with inherent sampling error). We go on to develop further analytic theory for the half-Gilbert model using stopping-set ideas once again, with some novel features. Using our theory, we obtain exact expressions for the first and second moments of the ray length in the half-Gilbert model. For all practical purposes, these results can be applied to the full-Gilbert model—as much better approximations than those provided by Mackisack and Miles (1996).

The vertices of the convex hull of a uniform sample from the interior of a convex polygon are known to be concentrated close to the vertices of the polygon. Furthermore, the remaining area of the polygon outside of the convex hull is concentrated close to the vertices of the polygon. In order to see what happens in a corner of the polygon given by two adjacent edges, we consider—in view of affine invariance—n points P1,…, Pn distributed independently and uniformly in the interior of the triangle with vertices (0, 1), (0, 0), and (1, 0). The number of vertices of the convex hull, which are close to the origin (0, 0), is then given by the number Ñn of points among P1,…, Pn, which are vertices of the convex hull of (0, 1), P1,…, Pn, and (1, 0). Correspondingly, D̃n is defined as the remaining area of the triangle outside of this convex hull. We derive exact (nonasymptotic) formulae for var Ñn and var . These formulae are in line with asymptotic distribution results in Groeneboom (1988), Nagaev and Khamdamov (1991), and Groeneboom (2012), as well as with recent results in Pardon (2011), (2012).

The almost-sure connectivity of the Euclidean minimal spanning forest MSF(X) on a homogeneous Poisson point process X ⊂ ℝd is an open problem for dimension d>2. We introduce a descending family of graphs (Gn)n≥2 that can be seen as approximations to the MSF in the sense that MSF(X)=∩n=2∞Gn(X). For n=2, one recovers the relative neighborhood graph or, in other words, the β-skeleton with β=2. We show that almost-sure connectivity of Gn(X) holds for all n≥2, all dimensions d≥2, and also point processes X more general than the homogeneous Poisson point process. In particular, we show that almost-sure connectivity holds if certain continuum percolation thresholds are strictly positive or, more generally, if almost surely X does not admit generalized descending chains.

Dimension reduction of multivariate data was developed by Y. Guan for point processes with Gaussian random fields as covariates. The generalization to fibre and surface processes is straightforward. In inverse regression methods, we suggest slicing based on geometrical marks. An investigation of the properties of this method is presented in simulation studies of random marked sets. In a refined model for dimension reduction, the second-order central subspace is analyzed in detail. A real data pattern is tested for independence of a covariate.

A stationary Poisson cylinder process in the d-dimensional Euclidean space is composed of a stationary Poisson process of k-flats (0 ≤ k ≤ d−1) which are dilated by independent and identically distributed random compact cylinder bases taken from the corresponding (d−k)-dimensional orthogonal complement. If the second moment of the (d−k)-volume of the typical cylinder base exists, we prove asymptotic normality of the d-volume of the union set of Poisson cylinders that covers an expanding star-shaped domain ϱ W as ϱ grows unboundedly. Due to the long-range dependencies within the union set of cylinders, the variance of its d-volume in ϱ W increases asymptotically proportional to the (d+k) th power of ϱ. To obtain the exact asymptotic behaviour of this variance, we need a distinction between discrete and continuous directional distributions of the typical k-flat. A corresponding central limit theorem for the surface content is stated at the end.

At each point of a Poisson point process of intensity λ in the hyperbolic plane, center a ball of bounded random radius. Consider the probability Pr that, from a fixed point, there is some direction in which one can reach distance r without hitting any ball. It is known (see Benjamini, Jonasson, Schramm and Tykesson (2009)) that if λ is strictly smaller than a critical intensity λgv thenPr does not go to 0 as r → ∞. The main result in this note shows that in the case λ=λgv, the probability of reaching a distance larger than r decays essentially polynomially, while if λ>λgv, the decay is exponential. We also extend these results to various related models and we finally obtain asymptotic results in several situations.

Given two correlated Brownian motions (Xt)t≥ 0 and (Yt)t≥ 0 with constant correlation coefficient, we give the upper and lower estimations of the probability ℙ(max0 ≤s≤tXs≥ a, max0 ≤s≤tYs≥ b) for any a,b,t > 0 through explicit formulae. Our strategy is to establish a new reflection principle for two correlated Brownian motions, which can be viewed as an extension of the reflection principle for one-dimensional Brownian motion. Moreover, we also consider the nonexit probability for linear boundaries, i.e. ℙ (Xt ≤ at+c,Yt ≤ bt+d, 0≤ t≤T) for any constants a, b≥0 and c,d, T > 0.

Wicksell's classical corpuscle problem deals with the retrieval of the size distribution of spherical particles from planar sections. We discuss the problem in a local stereology framework. Each particle is assumed to contain a reference point and the individual particle is sampled with an isotropic random plane through this reference point. Both the size of the section profile and the position of the reference point inside the profile are recorded and used to recover the distribution of the corresponding particle parameters. Theoretical results concerning the relationship between the profile and particle parameters are discussed. We also discuss the unfolding of the arising integral equations, uniqueness issues, and the domain of attraction relations. We illustrate the approach by providing reconstructions from simulated data using numerical unfolding algorithms.

We study a multiclass Markovian queueing network with switchover across a set of many-server stations. New arrivals to each station follow a nonstationary Poisson process. Each job waiting in queue may, after some exponentially distributed patience time, switch over to another station or leave the network following a probabilistic and state-dependent mechanism. We analyze the performance of such networks under the many-server heavy-traffic limiting regimes, including the critically loaded quality-and-efficiency-driven (QED) regime, and the overloaded efficiency-driven (ED) regime. We also study the limits corresponding to mixing the underloaded quality-driven (QD) regime with the QED and ED regimes. We establish fluid and diffusion limits of the queue-length processes in all regimes. The fluid limits are characterized by ordinary differential equations. The diffusion limits are characterized by stochastic differential equations, with a piecewise-linear drift term and a constant (QED) or time-varying (ED) covariance matrix. We investigate the load balancing effect of switchover in the mixed regimes, demonstrating the migration of workload from overloaded stations to underloaded stations and quantifying the load balancing impact of switchover probabilities.

We consider partial customer flexibility in service systems under two different designs. In the first design, flexible customers have their own queue and each server has its own queue of dedicated customers. Under this model, the problem is a scheduling problem and we show under various settings that the dedicated customers first (DCF) policy is optimal. In the second design, flexible customers are not queued separately and must be routed to one of the server's dedicated queues upon arrival. We extend earlier results about the ‘join the smallest work (JSW)’ policy to systems with dedicated as well as flexible arrivals. We compare these models to a routeing model in which only the queue length is available in terms of both efficiency and fairness and argue that the overall best approach for call centers is JSW routeing. We also discuss how this can be implemented in call centers even when work is unknown.

We consider the bipartite matching model of customers and servers introduced by Caldentey, Kaplan and Weiss (2009). Customers and servers play symmetrical roles. There are finite sets C and S of customer and server classes, respectively. Time is discrete and at each time step one customer and one server arrive in the system according to a joint probability measure μ on C× S, independently of the past. Also, at each time step, pairs of matched customers and servers, if they exist, depart from the system. Authorized matchings are given by a fixed bipartite graph (C, S, E⊂ C × S). A matching policy is chosen, which decides how to match when there are several possibilities. Customers/servers that cannot be matched are stored in a buffer. The evolution of the model can be described by a discrete-time Markov chain. We study its stability under various admissible matching policies, including ML (match the longest), MS (match the shortest), FIFO (match the oldest), RANDOM (match uniformly), and PRIORITY. There exist natural necessary conditions for stability (independent of the matching policy) defining the maximal possible stability region. For some bipartite graphs, we prove that the stability region is indeed maximal for any admissible matching policy. For the ML policy, we prove that the stability region is maximal for any bipartite graph. For the MS and PRIORITY policies, we exhibit a bipartite graph with a non-maximal stability region.

We consider a piecewise-multilinear interpolation of a continuous random field on a d-dimensional cube. The approximation performance is measured using the integrated mean square error. Piecewise-multilinear interpolator is defined by N-field observations on a locations grid (or design). We investigate the class of locally stationary random fields whose local behavior is like a fractional Brownian field, in the mean square sense, and find the asymptotic approximation accuracy for a sequence of designs for large N. Moreover, for certain classes of continuous and continuously differentiable fields, we provide the upper bound for the approximation accuracy in the uniform mean square norm.

Motivated by a wide range of applications, we consider a development of Whittle's restless bandit model in which project activation requires a state-dependent amount of a key resource, which is assumed to be available at a constant rate. As many projects may be activated at each decision epoch as resource availability allows. We seek a policy for project activation within resource constraints which minimises an aggregate cost rate for the system. Project indices derived from a Lagrangian relaxation of the original problem exist provided the structural requirement of indexability is met. Verification of this property and derivation of the related indices is greatly simplified when the solution of the Lagrangian relaxation has a state monotone structure for each constituent project. We demonstrate that this is indeed the case for a wide range of bidirectional projects in which the project state tends to move in a different direction when it is activated from that in which it moves when passive. This is natural in many application domains in which activation of a project ameliorates its condition, which otherwise tends to deteriorate or deplete. In some cases the state monotonicity required is related to the structure of state transitions, while in others it is also related to the nature of costs. Two numerical studies demonstrate the value of the ideas for the construction of policies for dynamic resource allocation, most especially in contexts which involve a large number of projects.

In this paper we present novel results for discrete-time and Markovian continuous-time multitype branching processes. As a population develops, we are interested in the waiting time until a particular type of interest (such as an escape mutant) appears, and in how the distribution of individuals depends on whether this type has yet appeared. Specifically, both forward and backward equations for the distribution of type-specific population sizes over time, conditioned on the nonappearance of one or more particular types, are derived. In tandem, equations for the probability that one or more particular types have not yet appeared are also derived. Brief examples illustrate numerical methods and potential applications of these results in evolutionary biology and epidemiology.

The asymptotic behavior is studied for an interacting particle system that involves independent motion and random sampling. For a fixed sampling rate, the empirical process of the particle system converges to the Fleming-Viot process when the number of particles approaches ∞. If the sampling rate approaches 0 as the number of particles becomes large, the corresponding empirical process will converge to the deterministic flow of the motion. In the main results of this paper, we study the corresponding central limit theorems and large deviations. Both the Gaussian limits and the large deviations depend on the sampling scales explicitly.

A probabilistic cellular automaton (PCA) can be viewed as a Markov chain. The cells are updated synchronously and independently, according to a distribution depending on a finite neighborhood. We investigate the ergodicity of this Markov chain. A classical cellular automaton is a particular case of PCA. For a one-dimensional cellular automaton, we prove that ergodicity is equivalent to nilpotency, and is therefore undecidable. We then propose an efficient perfect sampling algorithm for the invariant measure of an ergodic PCA. Our algorithm does not assume any monotonicity property of the local rule. It is based on a bounding process which is shown to also be a PCA. Last, we focus on the PCA majority, whose asymptotic behavior is unknown, and perform numerical experiments using the perfect sampling procedure.

The pricing of options in exponential Lévy models amounts to the computation of expectations of functionals of Lévy processes. In many situations, Monte Carlo methods are used. However, the simulation of a Lévy process with infinite Lévy measure generally requires either truncating or replacing the small jumps by a Brownian motion with the same variance. We will derive bounds for the errors generated by these two types of approximation.