Inaccuracy and information measures based on cumulative residual entropy are quite useful and have received considerable attention in many fields, such as statistics, probability, and reliability theory. In particular, many authors have studied cumulative residual inaccuracy between coherent systems based on system lifetimes. In a previous paper (Bueno and Balakrishnan, Prob. Eng. Inf. Sci. 36, 2022), we discussed a cumulative residual inaccuracy measure for coherent systems at component level, that is, based on the common, stochastically dependent component lifetimes observed under a non-homogeneous Poisson process. In this paper, using a point process martingale approach, we extend this concept to a cumulative residual inaccuracy measure between non-explosive point processes and then specialize the results to Markov occurrence times. If the processes satisfy the proportional risk hazard process property, then the measure determines the Markov chain uniquely. Several examples are presented, including birth-and-death processes and pure birth process, and then the results are applied to coherent systems at component level subject to Markov failure and repair processes.

We prove that for a discrete determinantal process the BK inequality occurs for increasing events generated by simple points. We also give some elementary but nonetheless appealing relationships between a discrete determinantal process and the well-known CS decomposition.

In this paper, we investigate pigeonhole statistics for the fractional parts of the sequence $\sqrt {n}$. Namely, we partition the unit circle $ \mathbb {T} = \mathbb {R}/\mathbb {Z}$ into N intervals and show that the proportion of intervals containing exactly j points of the sequence $(\sqrt {n} + \mathbb {Z})_{n=1}^N$ converges in the limit as $N \to \infty $. More generally, we investigate how the limiting distribution of the first $sN$ points of the sequence varies with the parameter $s \geq 0$. A natural way to examine this is via point processes—random measures on $[0,\infty )$ which represent the arrival times of the points of our sequence to a random interval from our partition. We show that the sequence of point processes we obtain converges in distribution and give an explicit description of the limiting process in terms of random affine unimodular lattices. Our work uses ergodic theory in the space of affine unimodular lattices, building upon work of Elkies and McMullen [Gaps in $\sqrt {n}$ mod 1 and ergodic theory. Duke Math. J. 123 (2004), 95–139]. We prove a generalisation of equidistribution of rational points on expanding horocycles in the modular surface, working instead on nonlinear horocycle sections.

We investigate the Hawkes processes on the positive real line exhibiting both self-excitation and inhibition. Each point of such a point process impacts its future intensity by the addition of a signed reproduction function. The case of a nonnegative reproduction function corresponds to self-excitation, and has been widely investigated in the literature. In particular, there exists a cluster representation of the Hawkes process which allows one to apply known results for Galton–Watson trees. We use renewal techniques to establish limit theorems for Hawkes processes that have reproduction functions which are signed and have bounded support. Notably, we prove exponential concentration inequalities, extending results of Reynaud-Bouret and Roy (2006) previously proven for nonnegative reproduction functions using a cluster representation no longer valid in our case. Importantly, we establish the existence of exponential moments for renewal times of M/G/$\infty$ queues which appear naturally in our problem. These results possess interest independent of the original problem.

In 2008, the Defense Advanced Research Project Agency commissioned a database known as the Integrated Crisis Early Warning System to serve as the foundation for models capable of detecting and predicting increases in political conflict worldwide. Such models, by signalling expected increases in political conflict, would help inform and prepare policymakers to react accordingly to conflict proliferation both domestically and internationally. Using data from the Integrated Crisis Early Warning System, we construct and test a self-exciting point process, or Hawkes process, model to describe and predict amounts of domestic, political conflict; we focus on Colombia and Venezuela as examples for this model. By comparing the accuracy of fitted models to the observed data, we find that we are able to closely describe occurrences of conflict in each country. Thus, using this model can allow policymakers to anticipate relative increases in the amount of domestic political conflict following major events.

The interest in retrial queueing systems mainly lies in their application to telephone systems. This paper studies multiserver retrial queueing systems with n servers. The arrival process is a quite general point process. An arriving customer occupies one of the free servers. If upon arrival all servers are busy, then the customer waits for his service in orbit, and after a random time retries in order to occupy a server. The orbit has one waiting space only, and an arriving customer, who finds all servers busy and the waiting space occupied, is lost from the system. Time intervals between possible retrials are assumed to have arbitrary distribution (the retrial scheme is explained more precisely in the paper). The paper provides analysis of this system. Specifically the paper studies the optimal number of servers to decrease the loss proportion to a given value. The representation obtained for the loss proportion enables us to solve the problem numerically. The algorithm for numerical solution includes effective simulation, which meets the challenge of a rare events problem in simulation.

The Palm version of a stationary random subset of a d-dimensional grid is contructed using the two-step change-of-origin and change-of-measure method. An elementary proof is given of the fact that the Palm version is characterized by point-stationarity (distributional invariance under bijective shifts of the origin from a point of the set to another point of the set).

Multivariate stochastic processes with Poisson marginals are of interest in insurance and finance; they can be used to model the joint behaviour of several claim arrival processes, for example. We discuss various methods for the construction of such models, with particular emphasis on the use of copulas. An important class of multivariate counting processes with Poisson marginals arises if the events of a background Poisson process with constant intensity are moved forward in time by a random amount and possibly deleted; here we think of the events of the background process as triggering later claims in different categories. We discuss structural aspects of these models, their dependence properties together with stochastic order aspects, and also some related computational issues. Various actuarial applications are indicated.

Empirical point processes of exceedances play an important role in extreme value theory, and their limiting behaviour has been extensively studied. Here, we provide explicit bounds on the accuracy of approximating an exceedance process by a compound Poisson or Poisson cluster process, in terms of a Wasserstein metric that is generally more suitable for the purpose than the total variation metric. The bounds only involve properties of the finite, empirical sequence that is under consideration, and not of any limiting process. The argument uses Bernstein blocks and Lindeberg's method of compositions.

In this paper we obtain the rate of convergence to equilibrium of a class of interacting marked point processes, introduced by Kerstan, in two different situations. Indeed, we prove the exponential and subexponential ergodicity of such a class of stochastic processes. Our results are an extension of the corresponding results of Brémaud, Nappo and Torrisi. The generality of the dynamics which we take into account allows the application to the so-called loss networks, and multivariate birth and death processes.

In this article we obtain rates of convergence to equilibrium of marked Hawkes processes in two situations. Firstly, the stationary process is the empty process, in which case we speak of the rate of extinction. Secondly, the stationary process is the unique stationary and nontrivial marked Hawkes process, in which case we speak of the rate of installation. The first situation models small epidemics, whereas the results in the second case are useful in deriving stopping rules for simulation algorithms of Hawkes processes with random marks.

With the aim of providing greater flexibility in developing and applying shot noise models, this paper studies shot noise on cluster point processes with both pointwise and cluster marks. For example, in financial modelling, responses to events in the financial market may occur in clusters, with random amplitudes including a ‘cluster component’ reflecting a degree of commonness among responses within a cluster. For such shot noise models, general formulae for the characteristic functional are developed and specialized to the case of Neyman-Scott clustering with cluster marks. For several general forms of response function, long range dependence of the corresponding equilibrium shot noise models is investigated. It is shown, for example, that long range dependence holds when the ‘structure component’ of the response function decays slowly enough, or when the response function has a finite random duration with a heavy tailed distribution.

In this article, we prove the existence of critical Hawkes point processes with a finite average intensity, under a heavy-tail condition for the fertility rate which is related to a long-range dependence property. Criticality means that the fertility rate integrates to 1, and corresponds to the usual critical branching process, and, in the context of Hawkes point processes with a finite average intensity, it is equivalent to the absence of ancestors. We also prove an ergodic decomposition result for stationary critical Hawkes point processes as a mixture of critical Hawkes point processes, and we give conditions for weak convergence to stationarity of critical Hawkes point processes.

The paper reviews the formulation of the linked stress release model for large scale seismicity together with aspects of its application. Using data from Taiwan for illustrative purposes, models can be selected and verified using tools that include Akaike's information criterion (AIC), numerical analysis, residual point processes and Monte Carlo simulation.

We define a class of anticipative flows on Poisson space and compute its Radon-Nikodym derivative. This result is applied to statistical testing in an anticipative queueing problem.

Recently Propp and Wilson [14] have proposed an algorithm, called coupling from the past (CFTP), which allows not only an approximate but perfect (i.e. exact) simulation of the stationary distribution of certain finite state space Markov chains. Perfect sampling using CFTP has been successfully extended to the context of point processes by, amongst other authors, Häggström et al. [5]. In [5] Gibbs sampling is applied to a bivariate point process, the penetrable spheres mixture model [19]. However, in general the running time of CFTP in terms of number of transitions is not independent of the state sampled. Thus an impatient user who aborts long runs may introduce a subtle bias, the user impatience bias. Fill [3] introduced an exact sampling algorithm for finite state space Markov chains which, in contrast to CFTP, is unbiased for user impatience. Fill's algorithm is a form of rejection sampling and similarly to CFTP requires sufficient monotonicity properties of the transition kernel used. We show how Fill's version of rejection sampling can be extended to an infinite state space context to produce an exact sample of the penetrable spheres mixture process and related models. Following [5] we use Gibbs sampling and make use of the partial order of the mixture model state space. Thus we construct an algorithm which protects against bias caused by user impatience and which delivers samples not only of the mixture model but also of the attractive area-interaction and the continuum random-cluster process.

The paper studies a model of repairable systems which is flexible enough to incorporate the standard imperfect repair and many other models from the literature. Palm stationarity of virtual ages, inter-failure times and degrees of repair is studied. A Loynes-type scheme and Harris recurrent Markov chains combined with coupling methods are used. Results on the weak total variation and moment convergences are obtained and illustrated by examples with IFR, DFR, heavy-tailed and light-tailed lifetime distributions. Some convergences obtained are monotone and/or at a geometric rate.

Stochastic comparison results for replacement policies are shown in this paper using the formalism of point processes theory. At each failure moment a repair is allowed. It is performed with a random degree of repair including (as special cases) perfect, minimal and imperfect repair models. Results for such repairable systems with schemes of planned replacements are also shown. The results are obtained by coupling methods for point processes.

Assessing the reliability of computer software has been an active area of research in computer science for the past twenty years. To date, well over a hundred probability models for software reliability have been proposed. These models have been motivated by seemingly unrelated arguments and have been the subject of active debate and discussion. In the meantime, the search for an ideal model continues to be pursued. The purpose of this paper is to point out that practically all the proposed models for software reliability are special cases of self-exciting point processes. This perspective unifies the very diverse approaches to modeling reliability growth and provides a common structure under which problems of software reliability can be discussed.

The so-called ‘Swiss Army formula', derived by Brémaud, seems to be a general purpose relation which includes all known relations of Palm calculus for stationary stochastic systems driven by point processes. The purpose of this article is to present a short, and rather intuitive, proof of the formula. The proof is based on the Ryll–Nardzewski definition of the Palm probability as a Radon-Nikodym derivative, which, in a stationary context, is equivalent to the Mecke definition.