Individuals in a population which grows according to the rules defining the simple branching process can mutate to novel allelic forms. We obtain limit theorems for the number of alleles present in any generation, the total number of alleles ever seen and the number of the generation containing the last mutation event.

In addition we define a notion of frequency spectrum for each generation as the expected number of alleles having a given number of representatives. As the generation number increases we prove the existence of a limiting notion of the frequency spectrum and discuss its upper tail behaviour. Our results here are incomplete and we make some conjectures which are supported by informal argument and specific examples.

For N = 1, 2, …, let {(XN (k), YN (k)), k = 0, 1, …} be a time-homogeneous Markov chain in . Suppose that, asymptotically as N → ∞, the ‘infinitesimal' covariances and means of XN([·/ε N]) are aij(x, y) and bi(x, y), and those of YN ([·/δ N]) are 0 and cl(x, y). Assume and limN→∞ε N/δ N = 0. Then, under a global asymptotic stability condition on dy/dt = c(x, y) or a related difference equation (and under some technical conditions), it is shown that (i) XN([·/ε N]) converges weakly to a diffusion process with coefficients aij(x, 0) and bi(x, 0) and (ii) YN([t/ε N]) → 0 in probability for every t > 0. The assumption in Ethier and Nagylaki (1980) that the processes are uniformly bounded is removed here.

The results are used to establish diffusion approximations of multiallelic one-locus stochastic models for mutation, selection, and random genetic drift in a finite, panmictic, diploid population. The emphasis is on rare, severely deleterious alleles. Models with multinomial sampling of genotypes in the monoecious, dioecious autosomal, and X-linked cases are analyzed, and an explicit formula for the stationary distribution of allelic frequencies is obtained.

We consider a finite-state-space, continuous-time Markov chain which is time reversible. The state space is partitioned into two sets, termed ‘open' and ‘closed', and it is only possible to observe which set the process is in. Further, short sojourns in either the open or closed sets of states will fail to be detected. We show that the dynamic stochastic properties of the observed process are completely described by an embedded Markov renewal process. The parameters of this Markov renewal process are obtained, allowing us to derive expressions for the moments and autocorrelation functions of successive sojourns in both the open and closed states. We illustrate the theory with a numerical study.

If {Xj, } is a sequence of i.i.d. random vectors in , when do there exist scaling constants bn > 0 such that the sequence of random sets converges almost surely in the space of compact subsets of to a limit set? A multivariate regular variation condition on a properly defined distribution tail guarantees the almost sure convergence but without certain regularity conditions surprises can occur. When a density exists, an exponential form of regular variation plus some regularity guarantees the convergence.

In this paper we consider a number of questions relating to the problem of determining quasi-stationary distributions for transient Markov processes. First we find conditions under which a measure or vector that is µ-invariant for a matrix of transition rates is also μ-invariant for the family of transition matrices of the minimal process it generates. These provide a means for determining whether or not the so-called stationary conditional quasi-stationary distribution exists in the λ-transient case. The process is not assumed to be regular, nor is it assumed to be uniform or irreducible. In deriving the invariance conditions we reveal a relationship between μ-invariance and the invariance of measures for related processes called the μ-reverse and the μ-dual processes. They play a role analogous to the time-reverse process which arises in the discussion of stationary distributions. Secondly we bring the related notions of detail-balance and reversibility into the realm of quasi-stationary processes. For example, if a process can be identified as being μ-reversible, the problem of determining quasi-stationary distributions is made much simpler. Finally, we consider some practical problems that emerge when calculating quasi-stationary distributions directly from the transition rates of the process. Our results are illustrated with reference to a variety of processes including examples of birth and death processes and the birth, death and catastrophe process.

The validity of the saddlepoint expansion evaluated at the point y is considered in the limit y tending to ∞. This is done for the expansions of the density and of the tail probability of the mean of n i.i.d. random variables and also for the expansion of the tail probability of a compound Poisson sum , where N is a Poisson random variable. We consider both general conditions that ensure the validity of the expansions and study the four classes of densities for X1 introduced in Daniels (1954).

A player starts at x in (-G, G) and attempts to leave the interval in a limited playing time. In the discrete-time problem, G is a positive integer and the position is described by a random walk starting at integer x, with mean increments zero, and variance increment chosen by the player from [0, 1] at each integer playing time. In the continuous-time problem, the player's position is described by an Ito diffusion process with infinitesimal mean parameter zero and infinitesimal diffusion parameter chosen by the player from [0, 1] at each time instant of play. To maximize the probability of leaving the interval (–G, G) in a limited playing time, the player should play boldly by always choosing largest possible variance increment in the discrete-time setting and largest possible diffusion parameter in the continuous-time setting, until the player leaves the interval. In the discrete-time setting, this result affirms a conjecture of Spencer. In the continuous-time setting, the value function of play is identified.

This paper analyzes, derives efficient computational procedures and numerically investigates the following fluid model which is of interest in manufacturing and communications: m producing machines supply a buffer, n consuming machines feed off it. Each machine independently alternates between exponentially distributed random periods in the ‘in service' and ‘failed' states. Producers/consumers have their own failure/repair rates and working capacities. When the buffer is either full or empty some of the machines in service are not utilized to capacity; otherwise they are fully utilized. Our main result is for the state distribution of the Markovian system in equilibrium which is the solution of a system of differential equations. The spectral expansion for its solution is obtained. Two important decompositions are obtained: the eigenvectors have the Kronecker-product form in lower-dimensional vectors; the characteristic polynomial is factored with each factor an explicitly given polynomial of degree at most 4. All eigenvalues are real. For each of various cases of the model, a system of linear equations is derived from the boundary conditions; their solution complete the spectral expansion. The count in operations of the entire procedure is O(m3n3): independence from buffer size exemplifies an important attraction of fluid models. Computations have revealed several interesting features, such as the benefit of small machines and the inelasticity of production rate to inventory. We also give results on the eigenvalues of a more general fluid model, reversible Markov drift processes.

Markov trials are a sequence of dependent trials with two outcomes, success and failure, which are the states of a Markov chain. The distribution of the number of successes in n Markov trials and the first-passage time for a specified number of successes are obtained using an augmented Markov chain model.

The concept of an extremal index, which is a measure of local dependence amongst the exceedances over a high threshold by a stationary sequence, has a natural interpretation as the reciprocal of mean cluster size. We exhibit a counterexample which shows that this interpretation is not necessarily correct.

The main characteristics of the time-nonhomogeneous Johnson–Mehl tessellation with specified nucleation intensity α (t) = αtß, β > −1 and α > 0 being constants and t time, are investigated as functions of β.

In this paper we consider the observed lifetime of a multistate monotone system and ‘the critical lower set' which causes the system deterioration. Then, under suitable conditions, we identify the component lifetime distribution using a Newton–Kantorovic iterative procedure as in Meilijson (1981).

A consecutive k-within-m-out-of-n system consists of n identical and stochastically independent components arranged on a line. The system will fail if and only if within m consecutive components, there are at least k failures. Let Tn be the system's lifetime. Then, under quite general conditions we prove that there is a positive constant a such that the random variable n1/kaTn converges to a Weibull distribution as n →∞.