In this paper we introduce Lévy-driven Cox point processes (LCPs) as Cox point processes with driving intensity function Λ defined by a kernel smoothing of a Lévy basis (an independently scattered, infinitely divisible random measure). We also consider log Lévy-driven Cox point processes (LLCPs) with Λ equal to the exponential of such a kernel smoothing. Special cases are shot noise Cox processes, log Gaussian Cox processes, and log shot noise Cox processes. We study the theoretical properties of Lévy-based Cox processes, including moment properties described by nth-order product densities, mixing properties, specification of inhomogeneity, and spatio-temporal extensions.

A systematic study of random Laguerre tessellations, weighted generalisations of the well-known Voronoi tessellations, is presented. We prove that every normal tessellation with convex cells in dimension three and higher is a Laguerre tessellation. Tessellations generated by stationary marked Poisson processes are then studied in detail. For these tessellations, we obtain integral formulae for geometric characteristics and densities of the typical k-faces. We present a formula for the linear contact distribution function and prove various limit results for convergence of Laguerre to Poisson-Voronoi tessellations. The obtained integral formulae are subsequently evaluated numerically for the planar case, demonstrating their applicability for practical purposes.

Most metrics between finite point measures currently used in the literature have the flaw that they do not treat differing total masses in an adequate manner for applications. This paper introduces a new metric d̅1 that combines positional differences of points under a closest match with the relative difference in total mass in a way that fixes this flaw. A comprehensive collection of theoretical results about d̅1 and its induced Wasserstein metric d̅2 for point process distributions are given, including examples of useful d̅1-Lipschitz continuous functions, d̅2 upper bounds for the Poisson process approximation, and d̅2 upper and lower bounds between distributions of point processes of independent and identically distributed points. Furthermore, we present a statistical test for multiple point pattern data that demonstrates the potential of d̅1 in applications.

We study the problem of maximizing the long-run average growth of total wealth for a logarithmic utility function under the existence of fixed and proportional transaction costs. The market model consists of one riskless asset and d risky assets. Impulsive control theory is applied to this problem. We derive a quasivariational inequality (QVI) of ‘ergodic’ type and obtain a weak solution for the inequality. Using this solution, we obtain an optimal investment strategy to achieve the optimal growth.

We discuss some issues regarding the accuracy of a quantile-based estimation of risk capital. In this context, extreme value theory (EVT) emerges naturally. The paper sheds some further light on the ongoing discussion concerning the use of a semi-parametric approach like EVT and the use of specific parametric models such as the g-and-h. In particular, we discusses problems and pitfalls evolving from such parametric models when using EVT and highlight the importance of the underlying second-order tail behavior.

This paper is concerned with the finiteness and large-time behaviour of moments of the overshoot and undershoot of a high level, and of their moment generating functions (MGFs), for a Lévy process which drifts to -∞ almost surely. This provides information relevant to quantities associated with the ruin of an insurance risk process. Results of Klüppelberg, Kyprianou, and Maller (2004) and Doney and Kyprianou (2006) for asymptotic overshoot and undershoot distributions in the class of Lévy processes with convolution equivalent canonical measures are shown to have moment and MGF convergence extensions.

We study the evolution in time of the joint distribution of a pair of Feller processes, related by the fact that some random time ago they were identical, evolving as a single Feller process; from that time on, they began to evolve independently, conditional on a state at the time of split, according to the same Feller transition probabilities. Such processes are involved in the Fisher-Wright model: the distribution of the time counted backwards from the present to the time of split in the past is a function of deterministic but time-varying effective size 2N of the population from which the two processes are sampled. In terms of a corresponding family of Feller operators, assuming asymptotic stability or ergodicity of the process of mutation, we find the limit form of the distribution of such pairs of processes sampled from decaying, asymptotically constant, and growing populations. In the case where mutation is not asymptotically stable or ergodic, limit distributions are found for the distribution of relative differences.

We give a functional central limit theorem for spatial birth and death processes based on the representation of such processes as solutions of stochastic equations. For any bounded and integrable function in Euclidean space, we define a family of processes which is obtained by integrals of this function with respect to the centered and scaled spatial birth and death process with constant death rate. We prove that this family converges weakly to a Gaussian process as the scale parameter goes to infinity. We do not need the birth rates to have a finite range of interaction. Instead, we require that the birth rates have a range of interaction that decays polynomially. In order to show the convergence of the finite-dimensional distributions of the above processes, we extend Penrose's multivariate spatial central limit theorem. An example of the asymptotic normalities of the time-invariance estimators for the birth rates of spatial point processes is given.

We study the survival properties of inhomogeneous Galton-Watson processes. We determine the so-called branching number (which is the reciprocal of the critical value for percolation) for these random trees (conditioned on being infinite), which turns out to be an almost sure constant. We also shed some light on the way in which the survival probability varies between the generations. When we perform independent percolation on the family tree of an inhomogeneous Galton-Watson process, the result is essentially a family of inhomogeneous Galton-Watson processes, parameterized by the retention probability p. We provide growth rates, uniformly in p, of the percolation clusters, and also show uniform convergence of the survival probability from the nth level along subsequences. These results also establish, as a corollary, the supercritical continuity of the percolation function. Some of our results are generalizations of results in Lyons (1992).

We consider a piecewise-deterministic Markov process (Xt) governed by a jump intensity function, a rate function that determines the behaviour between jumps, and a stochastic kernel describing the conditional distribution of jump sizes. The paper deals with the point process of upcrossings of some level b by (Xt). We prove a version of Rice's formula relating the stationary density of (Xt) to level crossing intensities and show that, for a wide class of processes (Xt), as b → ∞, the scaled point process where ν+(b) denotes the intensity of upcrossings of b, converges weakly to a geometrically compound Poisson process.

We consider the M/M/∞ queue with m primary servers and infinitely many secondary servers. All the servers are numbered and ordered. An arriving customer takes the lowest available server. We define the wasted spaces as the difference between the highest numbered occupied server and the total number of occupied servers. Letting ρ = λ0/μ be the ratio of arrival to service rates, we study the probability distribution of the wasted spaces asymptotically for ρ → ∞. We also give some numerical results and the tail behavior for ρ = O(1).

We analyze asymptotically a differential-difference equation that arises in a Markov-modulated fluid model. We use singular perturbation methods to analyze the problem with appropriate scalings of the two state variables. In particular, the ray method and asymptotic matching are used.

In this paper a limit theorem is proved that establishes conditions under which the distribution of the difference between the size of the initial event in a random sequence, modeled as a finite point process, and the largest subsequent event approaches a limiting form independent of the size of the initial event. The underlying assumptions are that the sizes of the individual events follow an exponential distribution, that the expected total number of events increases exponentially with the size of the initial event, and that the structure of the sequence approximates that of a Poisson process. Particular cases to which the results apply include sequences of independent and identically distributed exponential variables, and the epidemic-type aftershock (ETAS) branching process model in the subcritical case. In all these cases the form of the limit distribution is shown to be that of a double exponential (type-I extreme value distribution). In sampling from a family of aftershock sequences, with possibly different underlying parameters, the limit distribution is a mixture of such double exponential distributions. The conditions for the simple limit to exist relate to the approximation of the distribution of the number of events by a Poisson distribution. One such condition requires the coefficient of variation (ratio of standard deviation to mean) of the number of events to converge to 0 as the mean increases. The results provide a statistical background to Båth's law in seismology, which asserts that in an aftershock sequence the magnitude of the main shock is commonly around 1.2 units higher than the magnitude of the largest aftershock.

A weighted graph G is a pair (V, ℰ) containing vertex set V and edge set ℰ, where each edge e ∈ ℰ is associated with a weight We. A subgraph of G is a forest if it has no cycles. All forests on the graph G form a probability space, where the probability of each forest is proportional to the product of the weights of its edges. This paper aims to simulate forests exactly from the target distribution. Methods based on coupling from the past (CFTP) and rejection sampling are presented. Comparisons of these methods are given theoretically and via simulation.