The interpretation of the ‘standard’ Palm version of a stationary random measure ξ is that it behaves like ξ conditioned on containing the origin in its mass. The interpretation of the ‘modified’ Palm version is that it behaves like ξ seen from a typical location in its mass. In this paper we shall focus on the modified Palm version, comparing it with the standard version in the transparent case of mixed biased coin tosses, and then establishing a limit theorem that motivates the above interpretation in the case of random measures on locally compact second countable Abelian groups possessing Følner averaging sets.

During the last decades, quite a number of interacting particle systems have been introduced and studied in the crossover area of mathematics and statistical physics. Some of these can be seen as simplistic models for opinion formation processes in groups of interacting people. In the model introduced by Deffuant et al. (2000), agents that are neighbors on a given network graph, randomly meet in pairs and approach a compromise if their current opinions do not differ by more than a given threshold value θ. We consider the two-sided infinite path ℤ as the underlying graph and extend existing models to a setting in which opinions are given by probability distributions. Similar to what has been shown for finite-dimensional opinions, we observe a dichotomy in the long-term behavior of the model, but only if the initial narrow mindedness of the agents is restricted.

This paper deals with long-range dependence of random measures on ℝd. By examples, it is demonstrated that one must be careful in order to define it consistently. Therefore, we define long-range dependence by a rather specific second-order condition and provide an equivalent formulation involving the asymptotic behaviour of the Bartlett spectrum near the origin. Then it is shown that the defining condition may be formulated less strictly when the additional isotropy assumption holds. Finally, we present an example of a long-range dependent random measure based on the 0-level excursion set of a Gaussian random field for which the corresponding spectral density and its asymptotics are explicitly derived.

Let ξ be a random measure on a locally compact second countable topological group, and let X be a random element in a measurable space on which the group acts. In the compact case we give a natural definition of the concept that the origin is a typical location for X in the mass of ξ, and prove that when this holds, the same is true on sets placed uniformly at random around the origin. This new result motivates an extension of the concept of typicality to the locally compact case where it coincides with the concept of mass-stationarity. We describe recent developments in Palm theory where these ideas play a central role.

Directionally convex ordering is a useful tool for comparing the dependence structure of random vectors, which also takes into account the variability of the marginal distributions. It can be extended to random fields by comparing all finite-dimensional distributions. Viewing locally finite measures as nonnegative fields of measure values indexed by the bounded Borel subsets of the space, in this paper we formulate and study directionally convex ordering of random measures on locally compact spaces. We show that the directionally convex order is preserved under some of the natural operations considered on random measures and point processes, such as deterministic displacement of points, independent superposition, and thinning, as well as independent, identically distributed marking. Further operations on Cox point processes such as position-dependent marking and displacement of points are shown to preserve the order. We also examine the impact of the directionally convex order on the second moment properties, in particular on clustering and on Palm distributions. Comparisons of Ripley's functions and pair correlation functions, as well as examples, seem to indicate that point processes higher in the directionally convex order cluster more. In our main result we show that nonnegative integral shot noise fields with respect to the directionally convex ordered random measures inherit this ordering from the measures. Numerous applications of this result are shown, in particular to comparison of various Cox processes and some performance measures of wireless networks, in both of which shot noise fields appear as key ingredients. We also mention a few pertinent open questions.

We consider the Voronoi tessellation based on a stationary Poisson process N in ℝd. We provide a complete and explicit description of the Palm distribution describing N as seen from a randomly chosen (typical) point on a k-face of the tessellation. In particular, we compute the joint distribution of the d−k+1 neighbours of the k-face containing the typical point. Using this result as well as a fundamental general relationship between Palm probabilities, we then derive some properties of the typical k-face and its neighbours. Generalizing recent results of Muche (2005), we finally provide the joint distribution of the typical edge (typical 1-face) and its neighbours.

This paper investigates new properties concerning the multifractal structure of a class of random self-similar measures. These measures include the well-known Mandelbrot multiplicative cascades, sometimes called independent random cascades. We evaluate the scale at which the multifractal structure of these measures becomes discernible. The value of this scale is obtained through what we call the growth speed in Hölder singularity sets of a Borel measure. This growth speed yields new information on the multifractal behavior of the rescaled copies involved in the structure of statistically self-similar measures. Our results are useful in understanding the multifractal nature of various heterogeneous jump processes.

We describe a class of self-similar random measure that generalizes the class of stable, completely random measure, or, in one dimension, the class of processes with stable independent increments. As with the stable processes, the realizations are purely atomic, but the masses of the atoms are not necessarily independent, but rather characterized by self-similar dependence relations. Indeed, the class can be described most effectively in terms of the point process on the product space for the locations and sizes (‘marksȉ) of the atoms. Then self-similarity reduces to an invariance relation (‘biscale invariance’) on the distribution of this marked point process. The condition can be satisfied when the marked point process is compound Poisson, corresponding to the nonnegative stable processes, but is by no means restricted to this case. An example is given which modifies and extends Ogata's epidemic-type aftershock sequence model for earthquake occurrence.

We first give the existence and uniqueness result and a comparison theorem for backward stochastic differential equations with Brownian motion and Poisson process as the noise source in stopping time (unbounded) duration. Then we obtain the existence and uniqueness result for fully coupled forward-backward stochastic differential equation with Brownian motion and Poisson process in stopping time (unbounded) duration. We also proved a comparison theorem for this kind of equation.

The traditional Dirichlet process is characterized by its distribution on a measurable partition of the state space - namely, the Dirichlet distribution. In this paper, we consider a generalization of the Dirichlet process and the family of multivariate distributions it induces, with particular attention to a special case where the multivariate density function is tractable.

The main purpose of this work is to study and apply generalized contact distributions of (inhomogeneous) Boolean models Z with values in the extended convex ring. Given a convex body L ⊂ ℝd and a gauge body B ⊂ ℝd, such a generalized contact distribution is the conditional distribution of the random vector (dB(L,Z),uB(L,Z),pB(L,Z),lB(L,Z)) given that Z∩L = ∅, where Z is a Boolean model, dB(L,Z) is the distance of L from Z with respect to B, pB(L,Z) is the boundary point in L realizing this distance (if it exists uniquely), uB(L,Z) is the corresponding boundary point of B (if it exists uniquely) and lB(L,·) may be taken from a large class of locally defined functionals. In particular, we pursue the question of the extent to which the spatial density and the grain distribution underlying an inhomogeneous Boolean model Z are determined by the generalized contact distributions of Z.

New metrics are introduced in the space of random measures and are applied, with various modifications of the contraction method, to prove existence and uniqueness results for self-similar random fractal measures. We obtain exponential convergence, both in distribution and almost surely, of an iterative sequence of random measures (defined by means of the scaling operator) to a unique self-similar random measure. The assumptions are quite weak, and correspond to similar conditions in the deterministic case.

The fixed mass case is handled in a direct way based on regularity properties of the metrics and the properties of a natural probability space. Proving convergence in the random mass case needs additional tools, such as a specially adapted choice of the space of random measures and of the space of probability distributions on measures, the introduction of reweighted sequences of random measures and a comparison technique.

A parametric family of completely random measures, which includes gamma random measures, positive stable random measures as well as inverse Gaussian measures, is defined. In order to develop models for clustered point patterns with dependencies between points, the family is used in a shot-noise construction as intensity measures for Cox processes. The resulting Cox processes are of Poisson cluster process type and include Poisson processes and ordinary Neyman-Scott processes.

We show characteristics of the completely random measures, illustrated by simulations, and derive moment and mixing properties for the shot-noise random measures. Finally statistical inference for shot-noise Cox processes is considered and some results on nearest-neighbour Markov properties are given.

The germ-grain model is defined as the union of independent identically distributed compact random sets (grains) shifted by points (germs) of a point process. The paper introduces a family of stationary random measures in ℝd generated by germ-grain models and defined by the sum of contributions of non-overlapping parts of the individual grains. The main result of the paper is the central limit theorem for these random measures, which holds for rather general independently marked germ-grain models, including those with non-Poisson distribution of germs and non-convex grains. It is shown that this construction of random measures includes those random measures obtained by positively extended intrinsic volumes. In the Poisson case it is possible to prove a central limit theorem under weaker assumptions by using approximations by m-dependent random fields. Applications to statistics of the Boolean model are also discussed. They include a standard way to derive limit theorems for estimators of the model parameters.

We consider a stationary germ-grain model Ξ with convex and compact grains and the distance r(x) from x ε ℝd to Ξ. For almost all points x ε ℝd there exists a unique point p(x) in the boundary of Ξ such that r(x) is the length of the vector x-p(x), which is called the spherical contact vector at x. In this paper we relate the distribution of the spherical contact vector to the times it takes a typical boundary point of Ξ to hit another grain if all grains start growing at the same time and at the same speed. The notion of a typical point is made precise by using the generalized curvature measures of Ξ. The result generalizes a well known formula for the Boolean model. Specific examples are discussed in detail.

For autoregressive time series with positive innovations which either have heavy right or left tails, linear programming parameter estimates of the autoregressive coefficients have good rates of convergence. However, the asymptotic distribution of the estimators depends heavily on the distribution of the process and thus cannot be used for inference. A bootstrap procedure circumvents this difficulty. We verify the validity of the bootstrap and also give some general comments on the bootstrapping of heavy tailed phenomena.

We consider a continuous polling system in heavy traffic. Using the relationship between such systems and age-dependent branching processes, we show that the steady-state number of waiting customers in heavy traffic has approximately a gamma distribution. Moreover, given their total number, the configuration of these customers is approximately deterministic.

The object studied in this paper is a pair (Φ, Y), where Φ is a random surface in and Y a random vector field on . The pair is jointly stationary, i.e. its distribution is invariant under translations. The vector field Y is smooth outside Φ but may have discontinuities on Φ. Gauss' divergence theorem is applied to derive a flow conservation law for Y. For this specializes to a well-known rate conservation law for point processes. As an application, relationships for the linear contact distribution of Φ are derived.

Mecke's formula is concerned with a stationary random measure and shift-invariant measure on a locally compact Abelian group , and relates integrations concerning them to each other. For , we generalize this to a pair of random measures which are jointly stationary. The resulting formula extends the so-called Swiss Army formula, which was recently obtained as a generalization for Little's formula. The generalized Mecke formula, which is called GMF, can be also viewed as a generalization of the stationary version of H = λG. Under the stationary and ergodic assumptions, we apply it to derive many sample path formulas which have been known as extensions of H = λG. This will make clear what kinds of probabilistic conditions are sufficient to get them. We also mention a further generalization of Mecke's formula.