In this part we extend and particularise results developed by the author in Part I (pp. 349–375) for a class of queueing systems which can be formulated as Markov renewal processes. We examine those models where the basic transition consists of only two types: ‘arrivals' and ‘departures'. The ‘arrival lobby' and ‘departure lobby' queue-length processes are shown, using the results of Part I to be Markov renewal. Whereas the initial study focused attention on the behaviour of the embedded discrete-time Markov chains, in this paper we examine, in detail, the embedded continuous-time semi-Markov processes. The limiting distributions of the queue-length processes in both continuous and discrete time are derived and interrelationships between them are examined in the case of continuous-time birth–death queues including the M/M/1/M and M/M/1 variants. Results for discrete-time birth–death queues are also derived.

We study a nonparametric method for estimating the boundary measure of a compact body G ⊂ ℝd (the boundary length when d = 2 and the surface area for d = 3) in the case when this measure agrees with the corresponding Minkowski content. The estimator we consider is closely related to the one proposed in Cuevas, Fraiman and Rodríguez-Casal (2007). Our method relies on two sets of random points, drawn inside and outside the set G, with different sampling intensities. Strong consistency and asymptotic normality are obtained under some shape hypotheses on the set G. Some applications and practical aspects are briefly discussed.

A difficulty in the application [5] of the method of polynomials as exposited by Galambos is investigated. The method, recast as the method of indicators, in a form due originally to Rényi [6], is applied to the situation of non-constant coefficients.

In this work we investigate under what circumstances the equilibrium distribution of the numbers of groups of various sizes in a certain M/G/k group-arrival group-departure loss system can be obtained in a closed product form.

In this paper we study a general class of population genetic models where the total population is divided into a number of subpopulations or types. Migration between subpopulations is fast. Extending the results of Nordborg and Krone (2002) and Sagitov and Jagers (2005), we prove, as the total population size N tends to ∞, weak convergence of the joint ancestry of a given sample of haploid individuals in the Skorokhod topology towards Kingman's coalescent with a constant change of time scale c. Our framework includes age-structured models, geographically structured models, and combinations thereof. We also allow each individual to have offspring in several subpopulations, with general dependency structures between the number of offspring of various types. As a byproduct, explicit expressions for the coalescent effective population size N/c are obtained.

Suppose X is a multidimensional diffusion process. Assume that at time zero the state of X is fully observed, but at time $T>0$ only linear combinations of its components are observed. That is, one only observes the vector $L X_T$ for a given matrix L. In this paper we show how samples from the conditioned process can be generated. The main contribution of this paper is to prove that guided proposals, introduced in [35], can be used in a unified way for both uniformly elliptic and hypo-elliptic diffusions, even when L is not the identity matrix. This is illustrated by excellent performance in two challenging cases: a partially observed twice-integrated diffusion with multiple wells and the partially observed FitzHugh–Nagumo model.

Let η = (η1,…,ηn) be a positive random vector. If its coordinates ηi and ηj are exchangeable, i.e. the distribution of η is invariant with respect to the swap πij of its ith and jth coordinates, then Ef(η) = Ef(πijη) for all integrable functions f. In this paper we study integrable random vectors that satisfy this identity for a particular family of functions f, namely those which can be written as the positive part of the scalar product 〈u, η〉 with varying weights u. In finance such functions represent payoffs from exchange options with η being the random part of price changes, while from the geometric point of view they determine the support function of the so-called zonoid of η. If the expected values of such payoffs are πij-invariant, we say that η is ij-swap-invariant. A full characterisation of the swap-invariance property and its relationship to the symmetries of expected payoffs of basket options are obtained. The first of these results relies on a characterisation theorem for integrable positive random vectors with equal zonoids. Particular attention is devoted to the case of asset prices driven by Lévy processes. Based on this, concrete semi-static hedging techniques for multi-asset barrier options, such as weighted barrier swap options, weighted barrier quanto-swap options, or certain weighted barrier spread options, are suggested.

Motivated by applications to a wide range of areas, including assemble-to-order systems, operations scheduling, healthcare systems, and the collaborative economy, we study a stochastic matching model on hypergraphs, extending the model of Mairesse and Moyal (J. Appl. Prob.53, 2016) to the case of hypergraphical (rather than graphical) matching structures. We address a discrete-event system under a random input of single items, simply using the system as an interface to be matched in groups of two or more. We primarily study the stability of this model, for various hypergraph geometries.

We study stationary processes given as solutions to stochastic differential equations driven by fractional Brownian motion. This rich class includes the fractional Ornstein-Uhlenbeck process and those processes that can be obtained from it by state space transformations. An explicit formula in terms of Euler's Γ-function describes the asymptotic behaviour of the covariance function of the fractional Ornstein-Uhlenbeck process near zero, which, by an application of Berman's condition, guarantees that this process is in the maximum domain of attraction of the Gumbel distribution. Necessary and sufficient conditions on the state space transforms are stated to classify the maximum domain of attraction of solutions to stochastic differential equations driven by fractional Brownian motion.

In this paper, we consider several stochastic models arising from environmental problems. First, we study pollution in a domain where undesired chemicals are deposited at random times and locations according to Poisson streams. The chemical concentration can be modeled by a linear stochastic partial differential equation (SPDE) which is solved by applying a general result. Various properties, especially the limit behavior of the pollution process, are discussed. Secondly, we consider the pollution problem when a tolerance level is imposed. The chemical concentration can still be modeled by a SPDE which is no longer linear. Its properties are investigated in this paper. When the leakage rate is positive, it is shown that the pollution process has an equilibrium state given by the deterministic model treated in [2]. Finally, the linear filtering problem is considered based on the data of several observation stations.

A central limit theorem for cumulative processes was first derived by Smith (1955). No remainder term was given. We use a different approach to obtain such a term here. The rate of convergence is the same as that in the central limit theorems for sequences of independent random variables.

In this paper we examine the extremal tail probabilities of moving sums in a marked Poisson random field. These sums are computed by adding up the weighted occurrences of events lying within a scanning set of fixed shape and size. We also provide an alternative representation of the constants of the asymptotic formulae in terms of the occupation measure of the conditional local random field at zero, and extend these representations to the constants of asymptotic tail probabilities of Gaussian random fields.

By using properties of up- and downcrossings of the sample functions of the workload process and of the attained waiting-time process for a G/G/1 queueing model, a direct proof of a theorem proved by Sakasegawa and Wolff is given.

Corrected random walk approximations to continuous-time optimal stopping boundaries for Brownian motion, first introduced by Chernoff and Petkau, have provided powerful computational tools in option pricing and sequential analysis. This paper develops the theory of these second-order approximations and describes some new applications.

In two earlier papers [6], [7] the properties of bivariate renewal processes and their associated two-dimensional renewal functions, H(x, y) were examined. By utilising the Fréchet bounds for joint distributions and the properties of univariate renewal processes, a collection of upper and lower bounds for H(x, y) are constructed. The evaluation of these bounds is carried out for the case of the family of bivariate Poisson processes. An interesting by-product of this investigation leads to a new inequality for the median of a Poisson random variable.

Discrete time-series models are commonly used to represent economic and physical data. In decision making and system control, the first-passage time and level-crossing probabilities of these processes against certain threshold levels are important quantities. In this paper, we apply an integral-equation approach together with the state-space representations of time-series models to evaluate level-crossing probabilities for the AR(p) and ARMA(1,1) models and the mean first passage time for AR(p) processes. We also extend Novikov's martingale approach to ARMA(p,q) processes. Numerical schemes are used to solve the integral equations for specific examples.

Consider a circle of circumference 1. Throw n points at random onto this circle and append to each of these points a clockwise arc of length s. The resulting random set is a union of a random number of connected components, each with specific size. Using tools designed by Steutel, we compute the joint distribution of the lengths of the connected components. Asymptotic results are presented when n goes to ∞ and s to 0 jointly according to different regimes.

Two Wiener-Hopf type factorization identities for multivariate distributions are introduced. Properties of associated stopping times are derived. The structure that produces one factorization also provides the unique solution of the Wiener-Hopf convolution equation on a convex cone in Rd. Some applications for multivariate storage and queueing systems are indicated. For a few cases explicit formulas are obtained for the transforms of the associated stopping times. A result of Kemperman is extended.

Results about stationary Poisson hyperplane processes and the induced hyperplane mosaics are extended to the case where, instead of stationarity, it is only assumed that the intensity measure has a (possibly continuous) density with respect to some translation-invariant measure. Intensities and quermass densities, which are constant in the stationary case, are then replaced by functions. In a similar way, the associated zonoid (Matheron's Steiner convex set) is generalized.

An invariant Φ of a tree T under a k-state Markov model, where the time parameter is identified with the edges of T, allows us to recognize whether data on N observed species can be associated with the N terminal vertices of T in the sense of having been generated on T rather than on any other tree with N terminals. The invariance is with respect to the (time) lengths associated with the edges of the tree. We propose a general method of finding invariants of a parametrized functional form. It involves calculating the probability f of all kN data possibilities for each of m edge-length configurations of T, then solving for the parameters using the m equations of form Φ (f) = 0. We apply this to the case of quadratic invariants for unrooted binary trees with four terminals, for all k, using the Jukes–Cantor type of Markov matrix. We report partial results on finding the smallest algebraically independent set of invariants.