We set up a model for electronic foreign-exchange markets, suggesting subordinators to represent sellers' and buyers' offers. Its analysis naturally leads to the study of level passage events. The classical level passage event concerns the joint law of the time, height, and jump size observed when a real-valued stochastic process first exceeds a given level h. We provide an up-to-date treatment in the case when this process is a subordinator, and extend these results to a multivariate setting.

Consider the single-server queue with an infinite buffer and a first-in–first-out discipline, either of type M/M/1 or Geom/Geom/1. Denote by 𝒜 the arrival process and by s the services. Assume the stability condition to be satisfied. Denote by 𝒟 the departure process in equilibrium and by r the time spent by the customers at the very back of the queue. We prove that (𝒟, r) has the same law as (𝒜, s), which is an extension of the classical Burke theorem. In fact, r can be viewed as the sequence of departures from a dual storage model. This duality between the two models also appears when studying the transient behaviour of a tandem by means of the Robinson–Schensted–Knuth algorithm: the first and last rows of the resulting semistandard Young tableau are respectively the last instant of departure from the queue and the total number of departures from the store.

The probability that two randomly selected phylogenetic trees of the same size are isomorphic is found to be asymptotic to a decreasing exponential modulated by a polynomial factor. The number of symmetrical nodes in a random phylogenetic tree of large size obeys a limiting Gaussian distribution, in the sense of both central and local limits. The probability that two random phylogenetic trees have the same number of symmetries asymptotically obeys an inverse square-root law. Precise estimates for these problems are obtained by methods of analytic combinatorics, involving bivariate generating functions, singularity analysis, and quasi-powers approximations.

An expression is derived for the proportion of quadrilaterals among the polygons formed by random lines in the plane. Its value is 0.381 466….

A classical theorem on the convergence to a diffusion of a sequence of branching processes will be proved in the case where the branching processes are modified to allow dependence on population size of the generating function governing reproduction. Some recent semigroup convergence theorems due to Kurtz (1969), (1975) provide the main tools.

The familiar problem of maximizing the probability of choosing the best from a group of N candidates, where N is known, is extended to the case of N unknown. An a priori distribution is assumed for N, and the case of a uniform distribution is examined. Let VN denote the probability of choosing the best from a group of at most N candidates, then it is shown that limN→∞VN = 2e–2.

In previous papers (1976), (1977) limit theorems were obtained for the classical Ising model in the absence of an external magnetic field, thereby providing a basis for asymptotic inference. The present paper extends these results to arbitrary external magnetic fields. Statistical inference for this model is important because its nearest-neighbour interactions provide a natural first approximation to spatial interaction among binary variables located on square lattices.

The most interesting behaviour occurs in zero field and at or beyond the critical point. In this case, the central limit result for nearest-neighbour interactions requires an unusual norming, the limiting variances may depend on the nature of the boundary conditions, and there cannot be any central limit result for external magnetic field. The implications of these phenomena for statistical inference are also discussed. In particular, the maximum likelihood estimator of magnetic field is not consistent. Rather it appears to have a non-trivial asymptotic distribution.

The distribution of the volume occupied by random spheres in a cube is considered, both when the number of spheres is fixed and when their centres form a Poisson field. The mean and variance are obtained and in the latter case the distribution is proved to converge to normality. The probability of complete coverage is also obtained heuristically.

The distribution of the convex hull of a random sample of d-dimensional variables is described by embedding the collection of convex sets into the space of continuous functions on the unit sphere. Weak convergence of the normalized convex hull of a circular Gaussian sample to a process with extreme-value marginal distributions is demonstrated. The proof shows that an underlying sequence of point processes converges to a Poisson point process and then applies the continuous mapping theorem. Several properties of the limit process are determined.

For point processes, a pair of points is isolated if each is the nearest neighbor of the other, and its degree of isolation is the relative distance to the next-nearest point. The frequency with which such pairs occur and the distribution of their degree of isolation are obtained for Poisson processes in ℝv. Similar results are obtained for some related models, incidentally solving a problem proposed by D. P. Shine.

We consider the following secretary problem: items ranked from 1 to n are randomly selected without replacement, one at a time, and to ‘win' is to stop at an item whose overall rank is less than or equal to s, given only the relative ranks of the items drawn so far. Our method of analysis is based on the existence of an imbedded Markov chain and uses the technique of backwards induction. In principal the approach can be used to give exact results for any value of s; we do the working for s = 3. We give exact results for the optimal strategy, the probability of success and the distribution of T, and the total number of draws when the optimal strategy is implemented. We also give some asymptotic results for these quantities as n → ∞.

Availability is an important characteristic of a system. Different types of availability are defined. For the case when a sequence of bivariate random variables of lifetime and repair time are i.i.d. certain properties have been established previously. In practice, however, we need to consider the situation where these bivariate random variables are independent but not identically distributed. Properties of two measures of availability for the i.i.d. case are extended to this more general case.

In this paper we consider phase-type distributions, their Laplace transforms which are rational functions and their representations which are finite-state Markov chains with an absorbing state. We first prove that, in any representation, the minimal number of states which are visited before absorption is equal to the difference of degree between denominator and numerator in the Laplace transform of the distribution. As an application, we prove that when the Laplace transform has a denominator with n real poles and a numerator of degree less than or equal to one the distribution has order n. We show that, in general, this result can be extended neither to the case where the numerator has degree two nor to the case of non-real poles.

Throughout this paper, we shall write ℒ(W) = ℒ(Z) to mean the random variables W and Z have the same distribution. The relation “ℒ(W) = ℒ(;Z)” reads “the law of W equals the law of Z”.

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.

In this paper, we study a class of bisexual Galton-Watson branching processes in which the law of offspring distribution is dependent on the population size. Under a suitable condition on the offspring distribution, we prove that the limit of mean growth-rate per mating unit exists. Based on this limit, we give a criterion to identify whether the process admits ultimate extinction with probability one.

Let Xn be a sequence of integrable real random variables, adapted to a filtration (Gn). Define Cn = √{(1 / n)∑k=1nXk − E(Xn+1 | Gn)} and Dn = √n{E(Xn+1 | Gn) − Z}, where Z is the almost-sure limit of E(Xn+1 | Gn) (assumed to exist). Conditions for (Cn, Dn) → N(0, U) x N(0, V) stably are given, where U and V are certain random variables. In particular, under such conditions, we obtain √n{(1 / n)∑k=1nX_k - Z} = Cn + Dn → N(0, U + V) stably. This central limit theorem has natural applications to Bayesian statistics and urn problems. The latter are investigated, by paying special attention to multicolor randomly reinforced urns.

A generalization to continuous time is given for a discrete-time model of a birth and death process in a random environment. Some important properties of this process in the continuous-time setting are stated and proved including instability and extinction conditions, and when suitable absorbing barriers have been defined, methods are given for the calculation of extinction probabilities and the expected duration of the process.

In this short note, we present a simple characterization of the increasing convex ordering on the set of probability distributions on ℝ. We show its usefulness by providing a very short proof of a comparison result for M/GI/1 queues due to Daley and Rolski, and obtained by completely different means.

The maximum entropy principle is used to model uncertainty by a maximum entropy distribution, subject to some appropriate linear constraints. We give an entropy concentration theorem (whose demonstration is based on large deviation techniques) which is a mathematical justification of this statistical modelling principle. Then we indicate how it can be used in artificial intelligence, and how relevant prior knowledge is provided by some classical descriptive statistical methods. It appears furthermore that the maximum entropy principle yields to a natural binding between descriptive methods and some statistical structures.