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.

Stationary expectations corresponding to long-run averages of additive functionals on level-dependent quasi-birth-and-death processes are considered. Special cases include long-run average costs or rewards, moments and cumulants of steady-state queueing network performance measures, and many others. We provide a matrix-analytic scheme for numerically computing such stationary expectations without explicitly computing the stationary distribution of the process, which yields an algorithm that is as quick as its counterparts for stationary distributions but requires far less computer storage. Specific problems arising in the case of infinite state spaces are discussed and the application of the algorithm is demonstrated by a queueing network example.

We consider a generalized telegraph process which follows an alternating renewal process and is subject to random jumps. More specifically, consider a particle at the origin of the real line at time t=0. Then it goes along two alternating velocities with opposite directions, and performs a random jump toward the alternating direction at each velocity reversal. We develop the distribution of the location of the particle at an arbitrary fixed time t, and study this distribution under the assumption of exponentially distributed alternating random times. The cases of jumps having exponential distributions with constant rates and with linearly increasing rates are treated in detail.

A continuously monitored system is considered, which is subject to accumulating deterioration modelled as a gamma process. The system fails when its degradation level exceeds a limit threshold. At failure, a delayed replacement is performed. To shorten the down period, a condition-based maintenance strategy is applied, with imperfect repair. Mimicking virtual age models used for recurrent events, imperfect repair actions are assumed to lower the system degradation through a first-order arithmetic reduction of age model. Under these assumptions, Markov renewal equations are obtained for several reliability indicators. Numerical examples illustrate the behaviour of the system.

Continuing the work in Bertoin (2011) we study the distribution of the maximal number X*k of offspring amongst all individuals in a critical Galton‒Watson process started with k ancestors, treating the case when the reproduction law has a regularly varying tail F̅ with index −α for α > 2 (and, hence, finite variance). We show that X*k suitably normalized converges in distribution to a Fréchet law with shape parameter α/2; this contrasts sharply with the case 1< α<2 when the variance is infinite. More generally, we obtain a weak limit theorem for the offspring sequence ranked in decreasing order, in terms of atoms of a certain doubly stochastic Poisson measure.

Consider random vectors formed by a finite number of independent groups of independent and identically distributed random variables, where those of the last group are stochastically smaller than those of the other groups. Conditions are given such that certain functions, defined as suitable means of supermodular functions of the random variables of the vectors, are supermodular or increasing directionally convex. Comparisons based on the increasing convex order of supermodular functions of such random vectors are also investigated. Applications of the above results are then provided in risk theory, queueing theory, and reliability theory, with reference to (i) net stop-loss reinsurance premiums of portfolios from different groups of insureds, (ii) closed cyclic multiclass Gordon-Newell queueing networks, and (iii) reliability of series systems formed by units selected from different batches.

We consider possibly nonlinear distributional fixed-point equations on weighted branching trees, which include the well-known linear branching recursion. In Jelenković and Olvera-Cravioto (2012), an implicit renewal theorem was developed that enables the characterization of the power-tail asymptotics of the solutions to many equations that fall into this category. In this paper we complement the analysis in our 2012 paper to provide the corresponding rate of convergence.

Conditional distributions for affine Markov processes are at the core of present (defaultable) bond pricing. There is, however, evidence that Markov processes may not be realistic models for short rates. Fractional Brownian motion (FBM) can be introduced by an integral representation with respect to standard Brownian motion. Using a simple prediction formula for the conditional expectation of an FBM and its Gaussianity, we derive the conditional distributions of FBM and related processes. We derive conditional distributions for fractional analogies of prominent affine processes, including important examples like fractional Ornstein–Uhlenbeck or fractional Cox–Ingersoll–Ross processes. As an application, we propose a fractional Vasicek bond market model and compare prices of zero-coupon bonds to those achieved in the classical Vasicek model.

The distributions of discrete, continuous and conditional multiple window scan statistics are studied. The finite Markov chain imbedding technique has been applied to obtain the distributions of fixed window scan statistics defined from a sequence of Bernoulli trials. In this manuscript the technique is extended to compute the distributions of multiple window scan statistics and the exact powers for multiple pulse and Markov dependent alternatives. An application in blood component quality monitoring is provided. Numerical results are also given to illustrate our theoretical results.

The goal of this paper is to prove a result conjectured in Föllmer and Schachermayer (2007) in a slightly more general form. Suppose that S is a continuous semimartingale and satisfies a large deviations estimate; this is a particular growth condition on the mean-variance tradeoff process of S. We show that S then allows asymptotic exponential arbitrage with exponentially decaying failure probability, which is a strong and quantitative form of long-term arbitrage. In contrast to Föllmer and Schachermayer (2007), our result does not assume that S is a diffusion, nor does it need any ergodicity assumption.

This paper builds a mixture representation of the reliability function of the conditional residual lifetime of a coherent system in terms of the reliability functions of conditional residual lifetimes of order statistics. Some stochastic ordering properties for the conditional residual lifetime of a coherent system with independent and identically distributed components are obtained, based on the stochastically ordered coefficient vectors.

In this paper, new conditions for the stability of V-geometrically ergodic Markov chains are introduced. The results are based on an extension of the standard perturbation theory formulated by Keller and Liverani. The continuity and higher regularity properties are investigated. As an illustration, an asymptotic expansion of the invariant probability measure for an autoregressive model with independent and identically distributed noises (with a nonstandard probability density function) is obtained.

In reliability a number of failure processes for repairable items are described by point processes, depending on the types of repairs being performed on failures of items. In this paper we describe the failure processes of repairable items from heterogeneous populations and study the stochastic predictions of future processes which utilize the failure/repair history. Two types of repair processes, perfect and minimal repair processes, will be considered. The results will be derived under a general stochastic formulation/setting. Applications of the obtained results to many different areas will be discussed and, specifically, some reliability applications will be illustrated in detail.

We study the two-sided exit problem of a time-homogeneous diffusion process with tax payments of loss-carry-forward type and obtain explicit formulae for the Laplace transforms associated with the two-sided exit problem. The expected present value of tax payments until default, the two-sided exit probabilities, and, hence, the nondefault probability with the default threshold equal to the lower bound are solved as immediate corollaries. A sufficient and necessary condition for the tax identity in ruin theory is discovered.

This paper provides an analysis on a discrete version of the Ornstein-Uhlenbeck (OU) process which reflects the small discrete movements caused by the tick size effect. This discrete OU process is derived from matching the first two moments to those of the standard OU process in an infinitesimal sense. We discuss the distributional convergence from the discrete to the continuous processes, and show that the convergence speed is in the second order of the step (tick) size. We also provide some analytical results for the proposed discrete OU process itself, including the closed-form formula of the moment generating function and a full characterisation of the steady state distribution. These results enable us to examine the convergence order explicitly.

We construct a response adaptive design, described in terms of a two-color urn model, targeting fixed asymptotic allocations. We prove asymptotic results for the process of colors generated by the urn and for the process of its compositions. An application of the proposed urn model is presented in an estimation problem context.

We consider the tree-reduced path of a symmetric random walk on ℤd. It is interesting to ask about the number of turns Tn in the reduced path after n steps. This question arises from inverting the signatures of lattice paths: Tn gives an upper bound of the number of terms in the signature needed to reconstruct a ‘random’ lattice path with n steps. We show that, when n is large, the mean and variance of Tn in the asymptotic expansion have the same order as n, while the lower-order terms are O(1). We also obtain limit theorems for Tn, including the large deviations principle, central limit theorem, and invariance principle. Similar techniques apply to other finite patterns in a lattice path.

The problem of constructing an optimal coadapted coupling for a pair of symmetric random walks on Z2d was considered by Connor and Jacka (2008), and the existence of a coupling which is stochastically fastest in the class of all such coadapted couplings was demonstrated. In this paper we show how to generalise this construction to an optimal coadapted coupling for the continuous-time symmetric random walk on Knd, where Kn is the complete graph with n vertices. Moreover, we show that although this coupling is not maximal for any n (i.e. it does not achieve equality in the coupling inequality), it does tend to a maximal coupling as n → ∞.

We consider a branching population where individuals have independent and identically distributed (i.i.d.) life lengths (not necessarily exponential) and constant birth rates. We let Nt denote the population size at time t. We further assume that all individuals, at their birth times, are equipped with independent exponential clocks with parameter δ. We are interested in the genealogical tree stopped at the first time T when one of these clocks rings. This question has applications in epidemiology, population genetics, ecology, and queueing theory. We show that, conditional on {T<∞}, the joint law of (Nt, T, X(T)), where X(T) is the jumping contour process of the tree truncated at time T, is equal to that of (M, -IM, Y′M) conditional on {M≠0}. Here M+1 is the number of visits of 0, before some single, independent exponential clock e with parameter δ rings, by some specified Lévy process Y without negative jumps reflected below its supremum; IM is the infimum of the path YM, which in turn is defined as Y killed at its last visit of 0 before e; and Y′M is the Vervaat transform of YM. This identity yields an explanation for the geometric distribution of NT (see Kitaev (1993) and Trapman and Bootsma (2009)) and has numerous other applications. In particular, conditional on {NT=n}, and also on {NT=n,T<a}, the ages and residual lifetimes of the n alive individuals at time T are i.i.d. and independent of n. We provide explicit formulae for this distribution and give a more general application to outbreaks of antibiotic-resistant bacteria in the hospital.

This paper is concerned with the class of distributions, continuous or discrete, whose shape is monotone of finite integer order t. A characterization is presented as a mixture of a minimum of t independent uniform distributions. Then, a comparison of t-monotone distributions is made using the s-convex stochastic orders. A link is also pointed out with an alternative approach to monotonicity based on a stationary-excess operator. Finally, the monotonicity property is exploited to reinforce the classical Markov and Lyapunov inequalities. The results are illustrated by several applications to insurance.