We give necessary and sufficient conditions for various results connecting the tail behaviour of a distribution with that of its right Wiener–Hopf factor.

In a previous paper, [1] it was shown that a solution, f(x) will exist for the optimal inventory equation (where f(y − z) = f(0), y < z) provided:

1. 1. g(x) ≧ 0, x ≧ 0;

2. 2. 0 < a < 1;

3. 3. h(x) is monotonically nondecreasing, h(0) = 0;

4. 4. F is a distribution function on [0, ∞);

   (In [1], 1–4 were denoted collectively as (A).)

   and either

5. 5a. g(x) is continuous for all x ≧ 0;

6. 5b. limx→∞g(x) = ∞;

7. 5c h(x) is continuous for all x > 0 (Theorem 2 of [1]);

   or

8. 6. g(x) is uniformly continuous for all x ≧ 0 (Theorem 3 of [1]).

A model is studied in which each of several servers assembles finished products consisting of N different input items. Items of each type arrive independently at the assembly station and are grouped into N-tuples consisting of one item of each type. N-tuples are assembled into finished products by the servers on a first come-first-served basis. The model is analyzed by means of the theory of weak convergence, and functional limit theorems are obtained for appropriately normalized random functions induced by the queue size processes. The limits are expressed as functionals of multi-dimensional Wiener processes, with ordinary central limit theorems obtained as corollaries.

A technique is developed for the analysis of the asymptotic behaviour in the long run of queueing systems with two waiting lines. The generating function of the time-dependent joint queue-length distribution is obtained with the aid of the theory of boundary value problems of the Riemann–Hilbert type and by introducing a conformal mapping of the unit disk onto a given domain. In the asymptotic analysis an extensive use is made of theorems on the boundary behaviour of such conformal mappings.

Considering a random binary tree with n labelled leaves, we use a pruning procedure on this tree in order to construct a β(3/2,1/2)-coalescent process. We also use the continuous analogue of this construction, i.e. a pruning procedure on Aldous's continuum random tree, to construct a continuous state space process that has the same structure as the β-coalescent process up to some time change. These two constructions enable us to obtain results on the coalescent process, such as the asymptotics on the number of coalescent events or the law of the blocks involved in the last coalescent event.

A general method is developed to compute easy bounds of the weighted stationary probabilities for networks of queues which do not satisfy the standard product form. The bounds are obtained by constructing approximating reversible Markov chains. Thus, the bounds are insensitive with respect to service-time distributions. A special representation, called the tree-form solution, of the stationary distribution is used to derive the bounds. The results are applied to an overflow model.

In the theory of autoregressive model fitting it is of interest to know the asymptotic behaviour, for large sample size, of the coefficients fitted. A significant role is played in this connection by the moments of the norms of the inverse sample covariance matrices. We establish uniform boundedness results for these, first under generally weak conditions and then for the special case of (infinite order) processes. These in turn imply corresponding ergodic theorems for the matrices in question.

This paper considers the average optimality for a continuous-time Markov decision process in Borel state and action spaces, and with an arbitrarily unbounded nonnegative cost rate. The existence of a deterministic stationary optimal policy is proved under the conditions that allow the following; the controlled process can be explosive, the transition rates are weakly continuous, and the multifunction defining the admissible action spaces can be neither compact-valued nor upper semicontinuous.

For many practical situations in reliability engineering, components in the system are usually dependent since they generally work in a collaborative environment. In this paper we build sufficient conditions for comparing two coherent systems under different random environments in the sense of the usual stochastic, hazard rate, reversed hazard rate, and likelihood ratio orders. Applications and numerical examples are provided to illustrate all the theoretical results established here.

It is generally recognized that Alfred Lotka made the first application of standard Galton-Watson branching process theory to calculate an extinction probability in a specific population (using asexual reproduction). This note applies bisexual Galton-Watson branching process theory to the calculation of an extinction probability from Lotka's data, yielding a somewhat higher value.

Let {X(t), N(t)}, – ∞< t <∞, be a stationary bivariate stochastic process where X(t) is an ordinary time series and N(t) is an orderly point process counting the number of points in (0, t]. Suppose values of {X(t), N(t)} are available for 0< t ≦T and let σ1, σ2, ···, σ N(T) denote the jump points of N(t) in (0, T]. For |v| < T, define mT12(υ)=∑ X (σj+υ)/T and μT12(υ)=∑ X (σj+υ)/∑1 where all summations are over indices j such that 0<σj, σj+υ≦T for some σj. The functions MT12(υ) and μT12(υ) are often useful in analyzing the covariation of the time series and point process. In this paper, we shall develop some statistical properties of the functions MT12(υ) and μT12(υ) and discuss some specific situations where it is useful to consider these functions.

The Samaniego signature is a relevant tool for studying the performance of a system whose component lifetimes are exchangeable. It is well known that the stochastic ordering of the signatures of two systems implies the same for the respective system lifetimes. We prove that the reverse claim is not true when the component lifetimes are independent and identically distributed. There exist small proportions of systems with stochastically ordered lifetimes whose signatures are not ordered. We present a simple procedure in order to check whether the system lifetimes are stochastically ordered even if their signatures are not comparable.

By extending the methods of Peligrad et al. (2014), we establish exact moderate and large deviation asymptotics for linear random fields with independent innovations. These results are useful for studying nonparametric regression with random field errors and strong limit theorems.

The spectral factorization problem, i.e. the problem of obtaining all possible MA representations of a process with given autocovariance function, is considered for univariate, d-periodic MA(1) (equivalently, 1-dependent in the second-order sense) processes. The solutions are provided explicitly, and their invertibility properties are investigated. A characterization, in terms of their autocovariance functions, of non-invertible d-periodic 1-dependent processes, extending to the periodic case the traditional unit root condition, is provided.

It is well known that the central limit theorem holds for partial sums of a stationary sequence (Xi) of m-dependent random variables with finite variance; however, the limit may be degenerate with variance 0 even if var(Xi) ≠ 0. We show that this happens only in the case when Xi – EXi = Yi – Yi–1 for an (m − 1)-dependent stationary sequence (Yi) with finite variance (a result implicit in earlier results), and give a version for block factors. This yields a simple criterion that is a sufficient condition for the limit not to be degenerate. Two applications to subtree counts in random trees are given.

Many prominent continuous-time stochastic volatility models exhibit certain functional relationships between price jumps and volatility jumps. We show that stochastic volatility models like the Ornstein–Uhlenbeck and other continuous-time CARMA models as well as continuous-time GARCH and EGARCH models all exhibit such functional relations. We investigate the asymptotic behaviour of certain functionals of price and volatility processes for discrete observations of the price process on a grid, which are relevant for estimation and testing problems.

The limit behaviour in probability of realised quadratic variation is discussed under a relatively simple ambit process setting. The relation of this to the underlying volatility/intermittency field is in focus, especially as concerns the question of no volatility/intermittency memory.

We consider a random walks system on ℤ in which each active particle performs a nearest-neighbor random walk and activates all inactive particles it encounters. The movement of an active particle stops when it reaches a certain number of jumps without activating any particle. We prove that if the process relies on efficient particles (i.e. those particles with a small probability of jumping to the left) being placed strategically on ℤ, then it might survive, having active particles at any time with positive probability. On the other hand, we may construct a process that dies out eventually almost surely, even if it relies on efficient particles. That is, we discuss what happens if particles are initially placed very far away from each other or if their probability of jumping to the right tends to 1 but not fast enough.

This paper provides tools for the study of the Dirichlet random walk in Rd. We compute explicitly, for a number of cases, the distribution of the random variable W using a form of Stieltjes transform of W instead of the Laplace transform, replacing the Bessel functions with hypergeometric functions. This enables us to simplify some existing results, in particular, some of the proofs by Le Caër (2010), (2011). We extend our results to the study of the limits of the Dirichlet random walk when the number of added terms goes to ∞, interpreting the results in terms of an integral by a Dirichlet process. We introduce the ideas of Dirichlet semigroups and Dirichlet infinite divisibility and characterize these infinite divisible distributions in the sense of Dirichlet when they are concentrated on the unit sphere of Rd.

Onwards from the mid-twentieth century, the stochastic filtering problem has caught the attention of thousands of mathematicians, engineers, statisticians, and computer scientists. Its applications span the whole spectrum of human endeavour, including satellite tracking, credit risk estimation, human genome analysis, and speech recognition. Stochastic filtering has engendered a surprising number of mathematical techniques for its treatment and has played an important role in the development of new research areas, including stochastic partial differential equations, stochastic geometry, rough paths theory, and Malliavin calculus. It also spearheaded research in areas of classical mathematics, such as Lie algebras, control theory, and information theory. The aim of this paper is to give a brief historical account of the subject concentrating on the continuous-time framework.