By previous work of Giordano and the author, ergodic actions of $\mathbf{Z}$ (and other discrete groups) are completely classified measure-theoretically by their dimension space, a construction analogous to the dimension group used in $\text{C}^{\ast }$-algebras and topological dynamics. Here we investigate how far from approximately transitive (AT) actions can be that derive from circulant (and related) matrices. It turns out not very: although non-AT actions can arise from this method of construction, under very modest additional conditions, approximate transitivity arises. KIn addition, if we drop the positivity requirement in the isomorphism of dimension spaces, then all these ergodic actions satisfy an analogue of AT. Many examples are provided.

In Weil (2001) formulae were proved for stationary Boolean models Z in ℝd with convex or polyconvex grains, which express the densities (specific mean values) of mixed volumes of Z in terms of related mean values of the underlying Poisson particle process X. These formulae were then used to show that in dimensions 2 and 3 the densities of mixed volumes of Z determine the intensity γ of X. For d = 4, a corresponding result was also stated, but the proof given was incomplete, since in the formula for the density of the Euler characteristic V̅0(Z) of Z a term $\overline V^{(0)}_{2,2}(X,X)$ was missing. This was pointed out in Goodey and Weil (2002), where it was also explained that a new decomposition result for mixed volumes and mixed translative functionals would be needed to complete the proof. Such a general decomposition result has recently been proved by Hug, Rataj, and Weil (2013), (2018) and is based on flag measures of the convex bodies involved. Here, we show that such flag representations not only lead to a correct derivation of the four-dimensional result, but even yield a corresponding uniqueness theorem in all dimensions. In the proof of the latter we make use of Alesker’s representation theorem for translation invariant valuations. We also discuss which shape information can be obtained in this way and comment on the situation in the nonstationary case.

The fractional nonhomogeneous Poisson process was introduced by a time change of the nonhomogeneous Poisson process with the inverse α-stable subordinator. We propose a similar definition for the (nonhomogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombe’s theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.

We consider an infinitely divisible random field in ℝd given as an integral of a kernel function with respect to a Lévy basis. Under mild regularity conditions, we derive central limit theorems for the moment estimators of the mean and the variogram of the field.

We present the first algorithm that samples maxn≥0{Sn − nα}, where Sn is a mean zero random walk, and nα with $\alpha \in ({1 \over 2},1)$ defines a nonlinear boundary. We show that our algorithm has finite expected running time. We also apply this algorithm to construct the first exact simulation method for the steady-state departure process of a GI/GI/∞ queue where the service time distribution has infinite mean.

In the literature on optimal stopping, the problem of maximizing the expected discounted reward over all stopping times has been explicitly solved for some special reward functions (including (x+)ν, (ex − K)+, (K − e− x)+, x ∈ ℝ, ν ∈ (0, ∞), and K > 0) under general random walks in discrete time and Lévy processes in continuous time (subject to mild integrability conditions). All such reward functions are continuous, increasing, and logconcave while the corresponding optimal stopping times are of threshold type (i.e. the solutions are one-sided). In this paper we show that all optimal stopping problems with increasing, logconcave, and right-continuous reward functions admit one-sided solutions for general random walks and Lévy processes, thereby generalizing the aforementioned results. We also investigate in detail the principle of smooth fit for Lévy processes when the reward function is increasing and logconcave.

We consider a generalised Vervaat perpetuity of the form X = Y1W1 +Y2W1W2 + · · ·, where $W_i \sim {\cal U}^{1/t}$ and (Yi)i≥0 is an independent and identically distributed sequence of random variables independent from (Wi)i≥0. Based on a distributional decomposition technique, we propose a novel method for exactly simulating the generalised Vervaat perpetuity. The general framework relies on the exact simulation of the truncated gamma process, which we develop using a marked renewal representation for its paths. Furthermore, a special case arises when Yi = 1, and X has the generalised Dickman distribution, for which we present an exact simulation algorithm using the marked renewal approach. In particular, this new algorithm is much faster than existing algorithms illustrated in Chi (2012), Cloud and Huber (2017), Devroye and Fawzi (2010), and Fill and Huber (2010), as well as being applicable to the general payments case. Examples and numerical analysis are provided to demonstrate the accuracy and effectiveness of our method.

We study the Cramér type moderate deviation for partial sums of random fields by applying the conjugate method. The results are applicable to the partial sums of linear random fields with short or long memory and to nonparametric regression with random field errors.

A spatio-temporal model of particle or star growth is defined, whereby new unit masses arrive sequentially in discrete time. These unit masses are referred to as candidate stars, which tend to arrive in mass-dense regions and then either form a new star or are absorbed by some neighbouring star of high mass. We analyse the system as time increases, and derive the asymptotic growth rate of the number of stars as well as the size of a randomly chosen star. We also prove that the size-biased mass distribution converges to a Poisson–Dirichlet distribution. This is achieved by embedding our model into a continuous-time Markov process, so that new stars arrive according to a marked Poisson process, with locations as marks, whereas existing stars grow as independent Yule processes. Our approach can be interpreted as a Hoppe-type urn scheme with a spatial structure. We discuss its relevance for and connection to models of population genetics, particle aggregation, image segmentation, epidemic spread, and random graphs with preferential attachment.

We prove existence and uniqueness of a stationary distribution and absolute regularity for nonlinear GARCH and INGARCH models of order (p, q). In contrast to previous work we impose, besides a geometric drift condition, only a semi-contractive condition which allows us to include models which would be ruled out by a fully contractive condition. This results in a subgeometric rather than the more usual geometric decay rate of the mixing coefficients. The proofs are heavily based on a coupling of two versions of the processes.

We consider, for t in the boundary of a Galton–Watson tree $(\partial \textsf{T})$, the covering number $(\textsf{N}_n(t))$ by the generation-n cylinder. For a suitable set I and sequence (sn), we almost surely establish the Hausdorff dimension of the set $\{ t \in \partial {\textsf{T}}:{{\textsf{N}}_n}(t) - nb \ {\sim} \ {s_n}\} $ for b ∈ I.

A relationally exchangeable structure is a random combinatorial structure whose law is invariant with respect to relabeling its relations, as opposed to its elements. Historically, exchangeable random set partitions have been the best known examples of relationally exchangeable structures, but the concept now arises more broadly when modeling interaction data in modern network analysis. Aside from exchangeable random partitions, instances of relational exchangeability include edge exchangeable random graphs and hypergraphs, path exchangeable processes, and a range of other network-like structures. We motivate the general theory of relational exchangeability, with special emphasis on the alternative perspective it provides and its benefits in certain applied probability problems. We then prove a de Finetti-type structure theorem for the general class of relationally exchangeable structures.

In this paper, we study limit behavior for a Markov-modulated binomial counting process, also called a binomial counting process under regime switching. Such a process naturally appears in the context of credit risk when multiple obligors are present. Markov-modulation takes place when the failure/default rate of each individual obligor depends on an underlying Markov chain. The limit behavior under consideration occurs when the number of obligors increases unboundedly, and/or by accelerating the modulating Markov process, called rapid switching. We establish diffusion approximations, obtained by application of (semi)martingale central limit theorems. Depending on the specific circumstances, different approximations are found.

$\mathbb{Z}^{d}$-extensions of probability-preserving dynamical systems are themselves dynamical systems preserving an infinite measure, and generalize random walks. Using the method of moments, we prove a generalized central limit theorem for additive functionals of the extension of integral zero, under spectral assumptions. As a corollary, we get the fact that Green–Kubo’s formula is invariant under induction. This allows us to relate the hitting probability of sites with the symmetrized potential kernel, giving an alternative proof and generalizing a theorem of Spitzer. Finally, this relation is used to improve, in turn, the assumptions of the generalized central limit theorem. Applications to Lorentz gases in finite horizon and to the geodesic flow on Abelian covers of compact manifolds of negative curvature are discussed.

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.

In this paper we present a set of results relating to the occupation time α(t) of a process X(·). The first set of results concerns exact characterizations of α(t), e.g. in terms of its transform up to an exponentially distributed epoch. In addition, we establish a central limit theorem (entailing that a centered and normalized version of α(t)∕t converges to a zero-mean normal random variable as t→∞) and the tail asymptotics of ℙ(α(t)∕t≥q). We apply our findings to spectrally positive Lévy processes reflected at the infimum and establish various new occupation time results for the corresponding model.

Bizarrely shaped voting districts are frequently lambasted as likely instances of gerrymandering. In order to systematically identify such instances, researchers have devised several tests for so-called geographic compactness (i.e. shape niceness). We demonstrate that under certain conditions, a party can gerrymander a competitive state into geographically compact districts to win an average of over 70% of the districts. Our results suggest that geometric features alone may fail to adequately combat partisan gerrymandering.

In this paper we give a new flavour to what Peter Jagers and his co-authors call `the path to extinction'. In a neutral population of constant size N, assume that each individual at time 0 carries a distinct type, or allele. Consider the joint dynamics of these N alleles, for example the dynamics of their respective frequencies and more plainly the nonincreasing process counting the number of alleles remaining by time t. Call this process the extinction process. We show that in the Moran model, the extinction process is distributed as the process counting (in backward time) the number of common ancestors to the whole population, also known as the block counting process of the N-Kingman coalescent. Stimulated by this result, we investigate whether it extends (i) to an identity between the frequencies of blocks in the Kingman coalescent and the frequencies of alleles in the extinction process, both evaluated at jump times, and (ii) to the general case of Λ-Fleming‒Viot processes.

We extend the work of Antunović et al. (2016) on competing types in preferential attachment models to include cases where the types have different fitnesses, which may be either multiplicative or additive. We show that, depending on the values of the parameters of the models, there are different possible limiting behaviours depending on the zeros of a certain function. In particular, we show the existence of choices of the parameters where one type is favoured both by having higher fitness and by the type of attachment mechanism, but the other type has a positive probability of dominating the network in the limit.

Let Xn(k) be the number of vertices at level k in a random recursive tree with n+1 vertices. We are interested in the asymptotic behavior of Xn(k) for intermediate levels k=kn satisfying kn→∞ and kn=o(logn) as n→∞. In particular, we prove weak convergence of finite-dimensional distributions for the process (Xn ([knu]))u>0, properly normalized and centered, as n→∞. The limit is a centered Gaussian process with covariance (u,v)↦(u+v)−1. One-dimensional distributional convergence of Xn(kn), properly normalized and centered, was obtained with the help of analytic tools by Fuchs et al. (2006). In contrast, our proofs, which are probabilistic in nature, exploit a connection of our model with certain Crump–Mode–Jagers branching processes.