A problem of regrinding and recycling worn train wheels leads to a Markov population process with distinctive properties, including a product-form equilibrium distribution. A convenient framework for analyzing this process is via the notion of dynamic reversal, a natural extension of ordinary (time) reversal. The dynamically reversed process is of the same type as the original process, which allows a simple derivation of some important properties. The process seems not to belong to any class of Markov processes for which stationary distributions are known.

We consider a model for the spread of an epidemic through a population divided into m groups, in which infectives move from group to group and infect only within their current group. For both deterministic and stochastic versions of this model, the effect on the total size of the epidemic of varying the speed with which infectives move between groups is considered. We also compare the distribution of the total size of this model with that of a suitably matched model in which infectives cannot move between groups, but are able to infect outside their own group.

Through the study of a simple embedded martingale we obtain an extension of the Kesten–Stigum theorem and prove a central limit theorem for controlled Galton-Watson processes.

In a branching process with random environments, the probability of ultimate extinction is a function of the environment sequence, and is therefore a random variable. Explicit results about the distribution of this random variable are difficult to obtain in general. Here we assume independent and identically distributed environments and use the special properties of fractional linear generating functions to derive some explicit distributions, which may be singular or absolutely continuous, depending on the values of certain parameters. We also consider briefly tail behaviour close to 1, and provide an extension to cases where probability generating functions are not fractional linear.

A particular random walk on the integers leads to a new, tractable Markov chain. Of the stationary probabilities, we discuss the existence, some analytic properties and a factorization which leads to an algorithmic procedure for their numerical computation. We also consider the positive recurrence of some variants which each call for different mathematical arguments. Analogous results are derived for a continuous version on the positive reals.

Exact and ordinary lumpability in finite Markov chains is considered. Both concepts naturally define an aggregation of the Markov chain yielding an aggregated chain that allows the exact determination of several stationary and transient results for the original chain. We show which quantities can be determined without an error from the aggregated process and describe methods to calculate bounds on the remaining results. Furthermore, the concept of lumpability is extended to near lumpability yielding approximative aggregation.

Let X be a Lévy process on the real line, and let Fc denote the generalized arcsine law on [0, 1] with parameter c. Then t−1 ⨍0tP0(Xs > 0) ds → c as t → ∞ is a necessary and sufficient condition for t—1 ⨍0t1{Xs>0}ds to converge in P0 law to Fc. Moreover, P0(Xt > 0) = c for all t > 0 is a necessary and sufficient condition for t—1 ⨍0t1{Xs>0}ds under P0 to have law Fc for all t > 0. We give an elementary proof of these results, and show how to derive Spitzer's theorem for random walks in a simple way from the Lévy process version.

We consider the problem of conditioning a non-explosive birth and death process to remain positive until time T, and consider weak convergence of this conditional process as T → ∞. By a suitable almost sure construction we prove weak convergence. The almost sure construction used is of independent interest but relies heavily on the strong monotonic properties of birth and death processes.

The diffusions on the shape and size-and-shape spaces induced by brownian motions on the pre-size-and-shape spaces have been investigated in several papers (cf.). We here address the dual problem: the character of the diffusions on the pre-shape and pre-size-and-shape spaces which induce brownian motions on the shape and size-and-shape spaces. In particular we show that the shape and size-and-shape spaces for k labelled points in ℝm are stochastically complete if k > m and obtain the heat kernels of certain diffusions which induce brownian motions on the size-and-shape spaces.

We derive two kinds of rate conservation laws for describing the time-dependent behavior of a process defined with a stationary marked point process and starting at time 0. These formulas are called TRCLs (time-dependent rate conservation laws). It is shown that TRCLs are useful to study the transient behaviors of risk and storage processes with stationary claim and supply processes and with a general premium and release rates, respectively. Detailed discussions are given for the severity for the risk process, and for the workload process of a single-server queue.

The expected number of upcrossings for a max-stable process is computed and compared with known results for stable processes. Asymptotically the formulas are of the same order.

The aim of this paper is to extend the existing theory of second-order self-similar processes as defined by Cox (1984) from the univariate case to higher dimensions. Multivariate self-similar processes defined in terms of second-order theory for stationary time series can be used as models for long-range dependent observations when the marginal observations are long-range dependent. An interesting question concerns the correlation structure within the processes when the marginal processes are correlated. We show that the self-similarity requirement, as defined in this article, implies a cross-correlation structure similar to that for the marginal processes. This occurs both in the time domain and in the frequency domain. This fact can be used to obtain generalized least squares estimates for the long-range dependence parameters. We discuss some difficulties concerning estimation based on simulations.

We consider two weaker partial orderings among non-negative random variables. We derive some characterizations of the equivalence of two life distributions under the weaker orderings. These results lead to the characterizations of the equality of two non-negative random vectors in distributions by moments of extremes and sample mean. As the application of these results, we get various characterizations of the exponential distribution among HNBUE and HNWUE life distribution classes. The main results of Bhattacharjee and Sethuraman (1990), and Basu and Kirmani (1986) are special cases of these more general results.

Suppose that a device is subjected to shocks governed by a counting process N = {N(t), t ≧0}. The probability that the device survives beyond time t is then H̄(t)=Σk=0∞Q̄ℙ[N(t)=k], where Q̄k is the probability of surviving k shocks. It is known that H is NBU if the interarrivals Uk, ∊ ℕ+, are independent and NBU, and Q̄k+j ≦ Q̄k· Q̄j holds whenever k, j ∊ ℕ. Similar results hold for the classes of the NBUE and HNBUE distributions. We show that some other ageing classes have similar properties.

Mullooly (1988) provides sufficient conditions under which the variance of a left-truncated, non-negative random variable will be greater than the variance of the original variable. We consider this problem for the class of exponential mixtures, and provide an explicit expression for the inflation in variance in terms of the mixing density.

The hazard rate ordering is an ordering for random variables which compares lifetimes with respect to their hazard rate functions. It is stronger than the usual stochastic order for random variables, yet is weaker than the likelihood ratio ordering. The hazard rate ordering is particularly useful in reliability theory and survival analysis, owing to the importance of the hazard rate function in these areas. In this paper earlier work on the hazard rate ordering is reviewed, and extensive new results related to coherent systems are derived. Initially we fix the form of a coherent structure and investigate the effect on the hazard rate function of the system when we switch the lifetimes of its components from the vector (T1, · ··, Tn) to the vector (T′1, · ··, T′n), where the hazard rate functions of the two vectors are assumed to be comparable in some sense. Although the hazard rate ordering is closed under the formation of series systems, we see that this is not the case for parallel systems even when the system is a two-component parallel system with exponentially distributed lifetimes. A positive result shows that for two-component parallel systems with proportional hazards (λ1r(t), λ2r(t))), the more diverse (λ1, λ2) is in the sense of majorization the stronger is the system in the hazard rate ordering. Unfortunately even this result does not extend to parallel systems with more than two components, demonstrating again the delicate nature of the hazard rate ordering.

The principal result of the paper concerns the hazard rate ordering for the lifetime of a k-out-of-n system. It is shown that if τ k|n is the lifetime of a k-out-of-n system, then τ k|n is greater than τ k+1|n in the hazard rate ordering for any k. This has an interesting interpretation in the language of order statistics. For independent (not necessarily identically distributed) lifetimes T1, · ··, Tn, we let Tk:n represent the kth order statistic (in increasing order). Then it is shown that Tk +1:n is greater than Tk:n in the hazard rate ordering for all k = 1, ···, n − 1. The result does not, however, extend to the stronger likelihood ratio order.

Let ℱ be a countable plane triangulation embedded in ℝ2 in such a way that no bounded region contains more than finitely many vertices, and let Pp be Bernoulli (p) product measure on the vertex set of ℱ. Let E be the event that a fixed vertex belongs to an infinite path whose vertices alternate states sequentially. It is shown that the AB percolation probability function θΑΒ (p) = Pp(E) is non-decreasing in p for 0 ≦ p ≦ ½. By symmetry, θ AΒ(p) is therefore unimodal on [0, 1]. This result partially verifies a conjecture due to Halley and stands in contrast to the examples of Łuczak and Wierman.

We suggest a new universal method of stochastic simulation, allowing us to generate rather efficiently random vectors with arbitrary densities in a connected open region or on its boundary. Our method belongs to the class of dynamic Monte Carlo procedures and is based on a special construction of a Markov chain on the boundary of the region. Its remarkable feature is that this chain admits a simple simulation, based on a universal (depending only on the dimensionality of the space) stochastic driver.

We study the problem of maximizing the probability of stopping at an object which is best in at least one of a given set of criteria, using only stopping rules based on the knowledge of whether the current object is relatively best in each of the criteria. The asymptotic results for the case of independent criteria are shown to hold in certain cases where the componentwise maxima are, pairwise, either asymptotically independent or asymptotically full dependent.

An example of the former is a random sample from a bivariate correlated normal distribution; thus our results settle a question posed recently by T. S. Ferguson.

By an argument which involves matching sample paths, some useful equations for the probability distribution of the fundamental period in the MAP/G/1 queue are derived with less calculational effort than in earlier proofs. It is further shown that analogous equations hold for the MAP/SM/1 queueing model. These results are then used to derive explicit formulas for the mean vectors of the number served during and the duration of the fundamental period.