A new coalescent is introduced to study the genealogy of a sample from the infinite-alleles model of population genetics. This coalescent also records the age ordering of alleles in the sample. The distribution of this process is found explicitly for the Moran model, and is shown to be robust for a wide class of reproductive schemes.

Properties of the ages themselves and the relationship between ages and class sizes then follow readily.

The paper is devoted to a systematic discussion of recently developed techniques for the study of weak convergence of sequences of stochastic processes. The methods described make essential use of the semimartingale structure of the processes. Sufficient conditions for tightness including the results of Rebolledo are derived. The techniques are applied to a special class of processes, namely the D-semimartingales. Applications to multitype branching processes are given.

A method is reviewed for proving weak convergence in a function-space setting when regular variation is a sufficient condition. Point processes and weak convergence techniques involving continuity arguments play a central role. The method is dimensionless and holds computations to a minimum. Many applications of the methods to processes derived from sums and maxima are given.

We study a rule of growing a sequence {tn } of finite subtrees of an infinite m-ary tree T. Independent copies {ω (n)} of a Bernoulli-type process ω on m letters are used to trace out a sequence of paths in T. The tree tn is obtained by cutting each , at the first node such that at most σ paths out of , pass through it. Denote by Hn the length of the longest path, hn the length of the shortest path, and Ln the length of the randomly chosen path in tn. It is shown that, in probability, Hn – logan = O(1), hn – logb (n/log n) = 0(1), (or hn – logb (n/log log n) = O(1)), and that is asymptotically normal. The parameters a, b, c depend on the distribution of ω and, in case of a, also on σ. These estimates describe respectively the worst, the best and the typical case behavior of a ‘trie’ search algorithm for a dictionary-type information retrieval system, with σ being the capacity of a page.

A triangle with vertices z 1, z 2 , z 3 in the complex plane may be denoted by a vector Z , Z = [z 1, z 2, z 3]t . From a sequence of independent and identically distributed 3×3 circulants {C j }∞ 1, we may generate from Z 1 the sequence of vectors or triangles {Z j }∞ 1, by the rule Z j = C j Z j–1 (j> 1), Z 1=Z . The ‘shape’ of a set of points, the simplest case being three points in the plane has been defined by Kendall (1984). We give several alternative, ab initio discussions of the shape of a triangle, and proofs of a limit theorem for shape of the triangles in the sequence {Z j }∞ 1. In Appendix A, the shape concept is applied to the zeros of a cubic polynomial. Appendix B contains some further remarks about shape. Appendix C uses the methods of this paper to give proofs of generalizations of two old theorems on triangles.

The spectral factorization problem was solved in Hallin (1984) for the class of (non-stationary) m-variate MA(q) stochastic processes, i.e. the class of second-order q-dependent processes. It was shown that such a process generally admits an infinite (mq(mq +1)/2-dimensional) family of possible MA(q) representations. The present paper deals with the invertibility properties and asymptotic behaviour of these MA(q) models, in connection with the problem of producing asymptotically efficient forecasts. Invertible and borderline non-invertible models are characterized (Theorems 3.1 and 3.2). A criterion is provided (Theorem 4.1) by which it can be checked whether a given MA model is a Wold–Cramér decomposition or not; and it is shown (Theorem 4.2) that, under mild conditions, almost every MA model is asymptotically identical with some Wold–Cramér decomposition. The forecasting problem is investigated in detail, and it is established that the relevant invertibility concept, with respect to asymptotic forecasting efficiency, is what we define as Granger-Andersen invertibility rather than the classical invertibility concept (Theorem 5.3). The properties of this new invertibility concept are studied and contrasted with those of its classical counterpart (Theorems 5.2 and 5.4). Numerical examples are also treated (Section 6), illustrating the fact that non-invertible models may provide asymptotically efficient forecasts, whereas invertible models, in some cases, may not. The mathematical tools throughout the paper are linear difference equations (Green's matrices, adjoint operators, dominated solutions, etc.), and a matrix generalization of continued fractions.

In this note we deal with the stochastic difference equation of the form Y n+1 = AnYn + Bn , n∊ℤ, where the sequence is assumed to be strictly stationary and ergodic. By means of simple arguments a unique stationary solution of this equation is constructed. The stability of the stationary solution is the second subject of investigation. It is shown that under some additional assumptions

Many queues and related stochastic models, and in particular those that have a matrix-geometric stationary probability vector, have steady-state queue-length densities that are asymptotically geometric. The graph of the asymptotic rate η of these densities as a function of the traffic intensity ρ is the caudal characteristic curve. This is an informative graph from which a number of qualitative inferences about the behavior of the queue may be drawn.

The caudal characteristic curve may be computed (by elementary algorithms) for several useful models for which a complete exact numerical solution is not practically feasible. These include queues with certain types of superimposed arrival processes and/or multiple non-exponential servers. The necessary theorems which lead to the algorithmic procedures as well as the interpretation of several numerical examples are discussed.

This paper considers a queueing system with two classes of customers and a single server, where the service policy is of threshold type. As soon as the amount of work required by the class 1 customers is greater than a fixed threshold, the class 1 customers get the server's attention; otherwise the class 2 customers have the priority. Service interruptions can occur for both classes of customers on the basis of the above description of the service mechanism, and in this case the service interruption discipline is preemptive resume priority (PRP). This model, which turns out to be a generalization of the PRP queueing system, has potential applications in computer systems and in communication networks. For Poisson inputs, exponential (arbitrary) servicetime distribution for class 1 (class 2) customers, we derive the Laplace–Stieltjes transform of the stationary joint distribution of the workload of the server, by reducing the analysis to the resolution of a boundary value problem. Explicit formulas are obtained.

Let X 1, · ··, Xn be exchangeable random variables with finite variance and two sequences of constants satisfying a 1≦···≦an , b 1≦···≦bn . Suppose that a ′ 1, ···, a ′ n is a rearrangement of a 1, ···, an and that g(x) is a non-decreasing function. Then

The purpose of this letter is to study a modified GI/G/1 queueing system in which the server becomes unavailable for (independent) random periods each time he is free. This problem was first studied by Gelenbe and Iasnogorodski [6] who obtained the stationary law of the waiting time of a customer. We construct a simple probabilistic model coupling a G//G/1 queue with an autonomous server (in Borovkov's terminology [1]) with a GI/G/1 queue of classical type having the same characteristics, to compare them stochastically. We prove that the waiting time is a Markov chain, using a renewal process property which has not previously been noted.