We consider the asymptotic behaviour of a random walk when it exits from a symmetric region of the form {(x, n) :|x| ≤ rn b } as r → ∞. In order to be sure that this actually occurs, we treat only the case where the power b lies in the interval [0,½), and we further assume a condition that prevents the probability of exiting at either boundary tending to 0. Under these restrictions we establish necessary and sufficient conditions for the pth moment of the overshoot to be O(r q ), and for the overshoot to be tight, as r → ∞.

A new treatment of second-order self-similarity and asymptotic self-similarity for stationary discrete time series is given, based on the fixed points of a renormalisation operator with normalisation factors which are not assumed to be power laws. A complete classification of fixed points is provided, consisting of the fractional noise and one other class. A convenient variance time function approach to process characterisation is used to exhibit large explicit families of processes asymptotic to particular fixed points. A natural, general definition of discrete long-range dependence is provided and contrasted with common alternatives. The closely related discrete form of regular variation is defined, its main properties given, and its connection to discrete self-similarity explained. Folkloric results on long-range dependence are proved or disproved rigorously.

In this paper, we propose a customer-arrival-based insurance risk model, in which customers' potential claims are described as independent and identically distributed heavy-tailed random variables and premiums are the same for each policy. We obtain some precise large deviation results for the prospective-loss process under a mild assumption on the random index (in our case, the customer-arrival process), which is much weaker than that in the literature.

A new class of models for inhomogeneous spatial point processes is introduced. These locally scaled point processes are modifications of homogeneous template point processes, having the property that regions with different intensities differ only by a scale factor. This is achieved by replacing volume measures used in the density with locally scaled analogues defined by a location-dependent scaling function. The new approach is particularly appealing for modelling inhomogeneous Markov point processes. Distance-interaction and shot noise weighted Markov point processes are discussed in detail. It is shown that the locally scaled versions are again Markov and that locally the Papangelou conditional intensity of the new process behaves like that of a global scaling of the homogeneous process. Approximations are suggested that simplify calculation of the density, for example, in simulation. For sequential point processes, an alternative and simpler definition of local scaling is proposed.

The aim of this paper is to investigate discrete approximations of the exponential functional of Brownian motion (which plays an important role in Asian options of financial mathematics) with the help of simple, symmetric random walks. In some applications the discrete model could be even more natural than the continuous one. The properties of the discrete exponential functional are rather different from the continuous one: typically its distribution is singular with respect to Lebesgue measure, all of its positive integer moments are finite and they characterize the distribution. On the other hand, using suitable random walk approximations to Brownian motion, the resulting discrete exponential functionals converge almost surely to the exponential functional of Brownian motion; hence their limit distribution is the same as in the continuous case, namely that of the reciprocal of a gamma random variable, and so is absolutely continuous with respect to Lebesgue measure. In this way, we also give a new and elementary proof of an earlier result by Dufresne and Yor.

Generalizing the standard frailty models of survival analysis, we propose to model frailty as a weighted Lévy process. Hence, the frailty of an individual is not a fixed quantity, but develops over time. Formulae for the population hazard and survival functions are derived. The power variance function Lévy process is a prominent example. In many cases, notably for compound Poisson processes, quasi-stationary distributions of survivors may arise. Quasi-stationarity implies limiting population hazard rates that are constant, in spite of the continual increase of the individual hazards. A brief discussion is given of the biological relevance of this finding.

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.

A known gamma-type result for the Poisson process states that certain domains defined through configuration of the points (or ‘particles’) of the process have volumes which are gamma distributed. By proving the corresponding sequential gamma-type result, we show that in some cases such a domain allows for decomposition into subdomains each having independent exponentially distributed volumes. We consider other examples—based on the Voronoi and Delaunay tessellations—where a natural decomposition does not produce subdomains with exponentially distributed volumes. A simple algorithm for the construction of a typical Voronoi flower arises in this work. In our theoretical development, we generalize the classical theorem of Slivnyak, relating it to the strong Markov property of the Poisson process and to a result of Mecke and Muche (1995). This new theorem has interest beyond the specific problems being considered here.

We define a class of tessellation models based on perturbing or deforming standard tessellations such as the Voronoi tessellation. We show how distributions over this class of ‘deformed’ tessellations can be used to define prior distributions for models based on tessellations, and how inference for such models can be carried out using Markov chain Monte Carlo methods; stability properties of the algorithms are investigated. Our approach applies not only to fixed dimension problems, but also to variable dimension problems, in which the number of cells in the tessellation is unknown. We illustrate our methods with two real examples. The first relates to reconstructing animal territories, represented by the individual cells of a tessellation, from observation of an inhomogeneous Poisson point process. The second example involves the analysis of an image of a cross-section through a sample of metal, with the tessellation modelling the micro-crystalline structure of the metal.

This paper is devoted to a study of the integral of the workload process of the single server queue, in particular during one busy period. Firstly, we find asymptotics of the area 𝒜 swept under the workload process W(t) during the busy period when the service time distribution has a regularly varying tail. We also investigate the case of a light-tailed service time distribution. Secondly, we consider the problem of obtaining an explicit expression for the distribution of 𝒜. In the general GI/G/1 case, we use a sequential approximation to find the Laplace—Stieltjes transform of 𝒜. In the M/M/1 case, this transform is obtained explicitly in terms of Whittaker functions. Thirdly, we consider moments of 𝒜 in the GI/G/1 queue. Finally, we show asymptotic normality of .

This paper introduces a new graph constructed from a point process. The idea is to connect a point with its nearest neighbour, then to the second nearest and continue this process until the point belongs to the interior of the convex hull of these nearest neighbours. The number of such neighbours is called the degree of a point. We derive the distribution of the degree of the typical point in a Poisson process, prove a central limit theorem for the sum of degrees, and propose an edge-corrected estimator of the distribution of the degree that is unbiased for a stationary Poisson process. Simulation studies show that this degree is a useful concept that allows the separation of clustering and repulsive behaviour of point processes.

Strings are generated by sequences of independent draws from a given alphabet. For a given pattern H of length l and an integer n ≥ l, our goal is to maximize the probability of stopping on the last appearance of the pattern H in a string of length n (if any), given that, if we choose to stop on an occurrence of H, we must do so right away. This contrasts with the goals of several other investigations on patterns in strings such as computing the expected occurrence time and the probability of finding exactly r patterns in a string of fixed length. Several motivations are given for our problem ranging from relatively simple best choice problems to more difficult stopping problems allowing for a variety of interesting applications. We solve this problem completely in the homogeneous case for uncorrelated patterns. However, several of these results extend immediately to the inhomogeneous case. In the homogeneous case, optimal success probabilities are shown to vary, depending on characteristics of the pattern, essentially between the value 1/e and a new asymptotic constant 0.619…. These results demonstrate a considerable difference between the true optimal success probability compared with what a first approximation by heuristic arguments would yield. It is interesting to see that the so-called odds algorithm which gives the optimal rule for l = 1 yields an excellent approximation of the optimal rule for any l. This is important for applications because the odds algorithm is very convenient. We finally give a detailed asymptotic analysis for the homogeneous case.

Distance measurements are useful tools in stochastic geometry. For a Boolean model Z in ℝd , the classical contact distribution functions allow the estimation of important geometric parameters of Z. In two previous papers, several types of generalized contact distributions have been investigated and applied to stationary and nonstationary Boolean models. Here, we consider random sets Z which are generated as the union sets of Poisson processes X of k-flats, k ∈ {0, …, d-1}, and study distances from a fixed point or a fixed convex body to Z. In addition, we also consider the distances from a given flat or a flag consisting of flats to the individual members of X and investigate the associated process of nearest points in the flats of X. In particular, we discuss to which extent the directional distribution of X is determined by this point process. Some of our results are presented for more general stationary processes of flats.

We consider random recursive fractals and prove fine results about their local behaviour. We show that for a class of random recursive fractals the usual multifractal spectrum is trivial in that all points have the same local dimension. However, by examining the local behaviour of the measure at typical points in the set, we establish the size of fine fluctuations in the measure. The results are proved using a large deviation principle for a class of general branching processes which extends the known large deviation estimates for the supercritical Galton-Watson process.

We consider a failure-prone system which operates in continuous time and is subject to condition monitoring at discrete time epochs. It is assumed that the state of the system evolves as a continuous-time Markov process with a finite state space. The observation process is stochastically related to the state process which is unobservable, except for the failure state. Combining the failure information and the information obtained from condition monitoring, and using the change of measure approach, we derive a general recursive filter, and, as special cases, we obtain recursive formulae for the state estimation and other quantities of interest. Up-dated parameter estimates are obtained using the EM algorithm. Some practical prediction problems are discussed and an illustrative example is given using a real dataset.

In order to increase the variety of feasible models for random stationary tessellations (mosaics), two operations acting on tessellations are studied: superposition and iteration (the latter is also referred to as nesting). The superposition of two planar tessellations is the superposition of the edges of the cells of both tessellations. The iteration of tessellations means that one tessellation is chosen as a ‘frame’ tessellation. The single cells of this ‘frame’ are simultaneously and independently subdivided by cut-outs of tessellations of an independent and identically distributed sequence of tessellations. In the present paper, we investigate the limits for sequences of tessellations that are generated by consecutive application of superposition or iteration respectively. Sequences of (renormalised) superpositions of stationary planar tessellations converge weakly to Poisson line tessellations. For consecutive iteration the notion of stability of distributions is adapted and necessary conditions are formulated for those tessellations which may occur as limits.

Some geometric properties of Lp spaces are studied which shed light on the prediction of infinite variance processes. In particular, a Pythagorean theorem for Lp is derived. Improved growth rates for the moving average parameters are obtained.

Generalizing Matérn's (1960) two hard-core processes, marked point processes are considered as models for systems of varying-sized, nonoverlapping convex grains. A Poisson point process is generated and grains are placed at the points. The grains are supposed to have varying sizes but the same shape as a fixed convex grain, with spheres as an important special case. The pattern is thinned so that no grains overlap.We consider the thinning probability of a ‘typical point’ under various thinning procedures, the volume fraction of the resulting system of grains, the relation between the intensity of the point processes before and after thinning, and the corresponding size distributions. The study is inspired by problems in material fatigue, where cracks are supposed to be initiated by large defects.

Among the disks centered at a typical particle of the two-dimensionalPoisson-Voronoi tessellation, let R m be the radius of the largest included within the polygonal cell associated with that particle and R M be the radius of the smallest containing that polygonal cell. In this article, we obtain the joint distribution of R m and R M. This result is derived from the covering properties of the circle due toStevens, Siegel and Holst. The same method works for studying the Crofton cell associated with the Poisson line process in the plane. The computation of the conditional probabilitiesP{R M ≥ r + s | R m = r}reveals the circular property of the Poisson-Voronoi typical cells(as well as the Crofton cells) having a ‘large’ in-disk.

In this paper, the class of periodic Ornstein-Uhlenbeck processes is defined. It is shown that periodic Ornstein-Uhlenbeck processes are stationary Markov random fields and the class of stationary distributions is characterized. In particular, any self-decomposable distribution is the stationary distribution of some periodic Ornstein-Uhlenbeck process. As examples, gamma periodic Ornstein-Uhlenbeck processes and Gaussian periodic Ornstein-Uhlenbeck processes are considered.