The neutral two-locus model in population genetics is reformulated as a measure-valued diffusion process and is shown under certain conditions to have a unique stationary distribution and be weakly ergodic. The limits of the process and its stationary distribution as the recombination parameter tends to infinity are found. Genealogies are incorporated into the model, and it is shown that a random sample of size n from the population at stationarity has a common ancestor.

In this paper, we refine the concept of Γ-variation up to second order, and we give a characterization of this type of asymptotic behaviour. We apply our results to obtain uniform rates of convergence in the weak convergence of renormalised sample maxima to the double exponential distribution. In a second application we derive a rate of convergence result for the Hill estimator.

We consider a time reversible, continuous time Markov chain on a finite state space. The state space is partitioned into two sets, termed open and closed, and it is only possible to observe whether the process is in an open or a closed state. Further, short sojourns in either the open or closed states fail to be detected. We consider the situation when the length of minimal detectable sojourns follows a negative exponential distribution with mean μ–1. We show that the probability density function of observed open sojourns takes the form , where n is the size of the state space. We present a thorough asymptotic analysis of fO(t) as μ tends to infinity. We discuss the relevance of our results to the modelling of single channel records. We illustrate the theory with a numerical example.

A sequence of random triangles is constructed by choosing successively the three vertices of one triangle at random in the interior of its predecessor. A way is found to prove that the shapes of these triangles converge, almost surely, to collinear shapes, thus closing a gap in one of the central arguments of Mannion [5]. The new approach is based on a representation of the triangle process by a sequence of products of i.i.d. random matrices. We succeed in calculating the corresponding Lyapounov exponent.

The shape of the nth in a sequence of random triangle shapes is (xn, yn). It was shown in [8] and [9] that yn → 0, almost surely, as n → ∞. In this paper we show that x„ converges in distribution, as n→∞, to a random variable x, and we find the probability distribution of x. The convergence in law of xn enables us to complete an argument used in one part of [9].

We consider sampling with replacement of equiprobable groups of a fixed size m from a finite population S. Given a subset A ⊂ S, the distributions of (a) the number of distinct elements of A in a sample of size k and (b) the sample size necessary to obtain at least say n elements of A are given. Neat formulas are given especially for the expected values of these, as well as of some related random variables. Further we derive an optimal strategy to collect all elements of S under the assumptions that sampling one group costs α monetary units and that it is possible to purchase the elements which are missing at the end of the sampling procedure at a price of β > α/m per element.

Making use of the integral equations given in [1], [2] and [3], the asymptotic behaviour of the first-passage time (FPT) p.d.f.'s through certain time-varying boundaries, including periodic boundaries, is determined for a class of one-dimensional diffusion processes with steady-state density. Sufficient conditions are given for the cases both of single and of pairs of asymptotically constant and asymptotically periodic boundaries, under which the FPT densities asymptotically exhibit an exponential behaviour. Explicit expressions are then worked out for the processes that can be obtained from the Ornstein–Uhlenbeck process by spatial transformations. Some new asymptotic results for the FPT density of the Wiener process are finally proved, together with a few miscellaneous results.

In this paper we analyse the motion of a particle P whose velocity is represented by a three-valued telegraph process. We prove that the probability law of the process describing the position of P is a solution of a third-order, linear, partial differential equation.

We obtain probability distributions of some generalised versions of the process of random signals, as well as other probabilistic features of the related process.

Finally, accelerated motions of P (where acceleration follows the classical telegraph process) are also analysed.

We analyze a stable GI/G/1 queue that starts operating at time t = 0 with N0 ≠ 0 customers. First, we analyze the time required for this queue to empty for the first time. Under the assumption that both the interarrival and the service time distributions are of the exponential type, we prove that , where λ and μ are the arrival and the service rates. Furthermore, assuming in addition that the interarrival time distribution is of the non-lattice type, we show that the settling time of the queue is essentially equal to N0/(μ –λ); that is, we prove that where is the total variation distance between the distribution of the number of customers in the system at time t and its steady-state distribution. Finally, we show that there is a similarity between the queue we analyze and a simple fluid model.

This note is concerned with a system of non-renewable components in parallel subjected to a common environment which is described by a real-valued external stochastic process. Given the environment process, the components are supposed to work independently, and the corresponding failure rates are all increasing (or decreasing) functions of the observed current state. It is then proved that, under these assumptions, the association of the external process implies the association of the component lifelengths. Connection with existing results is underlined.

A star-like circuit-switching network with multistage service is considered. Likelihood ratio ordering is used to show that there exists an optimal threshold for the mean number of messages processed through the network as a function of the input traffic intensity.