The inverse stochastic dominance of degree r is a stochastic order of interest in several branches of economics. We discuss it in depth, the central point being the characterization in terms of the weak r-majorization of the vectors of expected order statistics. The weak r-majorization (a notion introduced in the paper) is a natural extension of the classical (reverse) weak majorization of Hardy, Littlewood and Pòlya. This work also shows the equivalence between the continuous majorization (of higher order) and the discrete r-majorization. In particular, our results make it clear that the cases r = 1, 2 differ substantially from those with r ≥ 3, a fact observed earlier by Muliere and Scarsini (1989), among other authors. Motivated by this fact, we introduce new stochastic orderings, as well as new social inequality indices to compare the distribution of the wealth in two populations, which could be considered as natural extensions of the first two dominance rules and the S-Gini indices, respectively.

It is well understood that a supercritical continuous-state branching process (CSBP) is equal in law to a discrete continuous-time Galton–Watson process (the skeleton of prolific individuals) whose edges are dressed in a Poissonian way with immigration which initiates subcritical CSBPs (non-prolific mass). Equally well understood in the setting of CSBPs and superprocesses is the notion of a spine or immortal particle dressed in a Poissonian way with immigration which initiates copies of the original CSBP, which emerges when conditioning the process to survive eternally. In this article we revisit these notions for CSBPs and put them in a common framework using the well-established language of (coupled) stochastic differential equations (SDEs). In this way we are able to deal simultaneously with all types of CSBPs (supercritical, critical, and subcritical) as well as understanding how the skeletal representation becomes, in the sense of weak convergence, a spinal decomposition when conditioning on survival. We have two principal motivations. The first is to prepare the way to expand the SDE approach to the spatial setting of superprocesses, where recent results have increasingly sought the use of skeletal decompositions to transfer results from the branching particle setting to the setting of measure valued processes. The second is to provide a pathwise decomposition of CSBPs in the spirit of genealogical coding of CSBPs via Lévy excursions, albeit precisely where the aforesaid coding fails to work because the underlying CSBP is supercritical.

We study the impact of a random environment on lifetimes of coherent systems with dependent components. There are two combined sources of this dependence. One results from the dependence of the components of the coherent system operating in a deterministic environment and the other is due to dependence of components of the system sharing the same random environment. We provide different sets of sufficient conditions for the corresponding stochastic comparisons and consider various scenarios, namely, (i) two different (as a specific case, identical) coherent systems operate in the same random environment; (ii) two coherent systems operate in two different random environments; (iii) one of the coherent systems operates in a random environment and the other in a deterministic environment. Some examples are given to illustrate the proposed reasoning.

A population which develops as a linear birth process from one individual gives rise to a family tree of size N which shows the relationships of the N members to each other. If this tree is given, but with no a priori information of which point is the root (the original member) we seek to utilize our knowledge of the stochastic manner in which the tree grew to estimate this root. A maximum likelihood estimate is obtained, and the probabilities that this estimate is the true kth point, or a member of the true rth generation are obtained as functions of N. As N → ∞ these probabilities converge to non-zero values for any fixed k, r, which are surprisingly large—for example, when k is 1, the limiting value is 1 — loge2 ≈ .30685.

The multi-level Monte Carlo method proposed by Giles (2008) approximates the expectation of some functionals applied to a stochastic process with optimal order of convergence for the mean-square error. In this paper a modified multi-level Monte Carlo estimator is proposed with significantly reduced computational costs. As the main result, it is proved that the modified estimator reduces the computational costs asymptotically by a factor (p / α)2 if weak approximation methods of orders α and p are applied in the case of computational costs growing with the same order as the variances decay.

A dynamic model for a random network evolving in continuous time is defined, where new vertices are born and existing vertices may die. The fitness of a vertex is defined as the accumulated in-degree of the vertex and a new vertex is connected to an existing vertex with probability proportional to a function b of the fitness of the existing vertex. Furthermore, a vertex dies at a rate given by a function d of its fitness. Using results from the theory of general branching processes, an expression for the asymptotic empirical fitness distribution {pk} is derived and analyzed for a number of specific choices of b and d. When b(i) = i + α and d(i) = β, that is, linear preferential attachment for the newborn and random deaths, then pk ∼ k-(2+α). When b(i) = i + 1 and d(i) = β(i + 1), with β < 1, then pk ∼ (1 + β)−k, that is, if the death rate is also proportional to the fitness, then the power-law distribution is lost. Furthermore, when b(i) = i + 1 and d(i) = β(i + 1)γ, with β, γ < 1, then logpk ∼ -kγ, a stretched exponential distribution. The momentaneous in-degrees are also studied and simulations suggest that their behaviour is qualitatively similar to that of the fitnesses.

Recently Miyazawa and Taylor (1997) proposed a new class of queueing networks with batch arrival batch service and assemble-transfer features. In such networks customers arrive and are served in batches, and may change size when a batch transfers from one node to another. With the assumption of an additional arrival process at each node when it is empty, they obtain a simple product-form steady-state probability distribution, which is a (stochastic) upper bound for the original network. This paper shows that this class of network possesses a set of non-standard partial balance equations, and it is demonstrated that the condition of the additional arrival process introduced by Miyazawa and Taylor is there precisely to satisfy the partial balance equations, i.e. it is necessary and sufficient not only for having a product form solution, but also for the partial balance equations to hold.

By considering the extreme behavior of bivariate random vectors with a polar representation R(u(T), v(T)), it is commonly assumed that the radial component R and the angular component T are stochastically independent. We investigate how to relax this rigid independence assumption such that conditional limit theorems can still be deduced. For this purpose, we introduce a novel measure for the dependence structure and present convenient criteria for validity of limit theorems possessing a geometrical meaning. Thus, our results verify a stability of the available limit results, which is essential in applications where the independence of the polar components is not necessarily present or exactly fulfilled.

The symmetric rendezvous problem on a network Q asks how two players, forced to use the same mixed strategy, can minimize their expected meeting time, starting from a known initial distribution on the nodes of Q. This minimum is called the (symmetric) ‘rendezvous value’ of Q. Traditionally, the players are assumed to receive no information while playing the game. We consider the effect on rendezvous times of giving the players some information about past actions and chance moves, enabling each of them to apply Bayesian updates to improve his knowledge of the other's whereabouts. This technique can be used to give lower bounds on the rendezvous times of the original game (without any revealed information). We consider the case in which they are placed a known distance apart on the line graph Q (known as ‘symmetric rendezvous on the line’). Our approach is to concentrate on a general analysis of the effect of revelations, rather than compute the best bounds possible with our technique.

To study the limiting behaviour of the random running-time of the FIND algorithm, the so-called FIND process was introduced by Grübel and Rösler [1]. In this paper an approach for determining the nth moment function is presented. Applied to the second moment this provides an explicit expression for the variance.

Consider an inhomogeneous Poisson process X on [0, T] whose unknown intensity function ‘switches' from a lower function g∗ to an upper function h∗ at some unknown point θ∗. What is known are continuous bounding functions g and h such that g∗(t) ≤ g(t) ≤ h(t) ≤ h∗(t) for 0 ≤ t ≤ T. It is shown that on the basis of n observations of the process X the maximum likelihood estimate of θ∗ is consistent for n →∞, and also that converges in law and in pth moment to limits described in terms of the unknown functions g∗ and h∗.

Results connecting Yaglom's limit and Z1 are obtained. For example, if α > 1, E[Zα1] and converge or diverge together.

The algorithm for the transient solution for the denumerable state Markov process with an arbitrary initial distribution is given in this paper. The transient queue length distribution for a general Markovian queueing system can be obtained by this algorithm. As examples, some numerical results are presented.

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.

A proposal is given for estimating the home range of an animal based on sequential sightings. We assume the given sightings are independent, identically distributed random vectors X1,· ··, Xn whose common distribution has compact support. If are the polar coordinates of the sightings, then is a sup-measure and corresponds to the right endpoint of the distribution . The corresponding upper semi-continuous function l(θ) is the boundary of the home range. We give a consistent estimator for the boundary l and under the assumption that the distribution of R1 given is in the domain of attraction of an extreme value distribution with bounded support, we are able to give an approximate confidence region.

In this paper we consider optimal stopping problems for a general class of reward functions under matrix-exponential jump-diffusion processes. Given an American call-type reward function in this class, following the averaging problem approach (see, for example, Alili and Kyprianou (2005), Kyprianou and Surya (2005), Novikov and Shiryaev (2007), and Surya (2007)), we give an explicit formula for solutions of the corresponding averaging problem. Based on this explicit formula, we obtain the optimal level and the value function for American call-type optimal stopping problems.

Strong mixing is a condition which is often assumed to prove limit theorems for strictly stationary processes. Leadbetter's condition D(un) is used to prove limit theorems for maxima of stationary processes.

A sufficient condition for strong mixing to hold is given for the case where the process satisfies a pth-order Markov property. This condition can be easy to check for when p is small. This point is illustrated by two examples of first-order autoregressive processes.

The condition D(un) is shown to hold for any stationary Markov process.

In this paper we study continuous flow finite buffer systems with input rates modulated by Markov chains. Discrete event simulations are applied for estimating loss probabilities. The simulations are executed under a twisted version of the original probability measure (importance sampling). We present a simple rule for determining a new measure, then show that the new measure matches the ‘most likely' empirical measure that we expect from large deviations arguments, and finally prove optimality of the new measure.

This paper presents a model for random motion on binary trees. The symmetric model is defined first, and then a model with drift is given. Formulas for absorption probabilities and expected absorption times are presented in both cases.