An expression is found for the stationary density of the allele frequencies, in the infinitely-many alleles model. It is assumed that all alleles are neutral, that there is a constant mutation rate, and that the population is sufficiently large for the diffusion model to apply. Bounds on some moments are calculated from the density, and some applications are made to the problem of which allele is oldest in a population. A postulate made by Ewens (1972), concerning the distribution of allele numbers in a finite random sample from the neutral diffusion population, is shown to be correct.

We establish a sufficient condition for which the expected area under the trajectory of the carrier process is directly proportional to the expected number of removed carriers in the class of carrier-borne, right-shift, epidemic models studied by Severo (1969a). This result generalizes the previous work of Downton (1972) and Jerwood (1974) for some special cases of these models. We use the result to compute expected costs in the carrier-borne model due to Downton (1968) when it is unlikely that all the susceptibles will be infected. We conclude by showing that for the special case considered by Weiss (1965) this treatment of the expected cost is reasonable for populations with a large initial number of susceptibles.

A stochastic model for inter-plant competition in a single-species population is presented, in which the probability that a given plant will be eliminated is defined in terms of the sizes and relative spatial locations of rival plants. The properties of the model are then investigated and illustrations of particular cases given, with particular emphasis on the distribution of the sizes of surviving plants. Empirical comparisons are made between the properties of the model and some experimental results due to Ford (1975b). Some possible extensions to the basic model are suggested.

We investigate the effect of a particular mode of approximation by means of four examples of its use; in each case the model approximated is a Markov process with discrete states in continuous time.

The motion of a particle is investigated in the presence of a velocity-dependent random force assumed to be proportional to velocity. Two different possibilities are considered, namely, the presence and absence of random driving force. In the absence of random driving force, the velocity and displacement auto-correlation function are calculated. The probability distribution in velocity space is also evaluated. It is found that in the absence of intrinsic damping, the energy of the particle increases without limit. The condition for the energetic stability of the particle in the presence of random driving force is obtained. The Fokker-Planck equation for the probability distribution in velocity space is derived from the stochastic Liouville equation for delta-correlated velocity-dependent random force.

Particles enter a finite-state system and move according to independent sample paths from a semi-Markov process. Strong limit theorems are developed for the ratio of the flow of particles from states i to j and the flow out of When the cumulative arrival of particles into the system up to time t, A (t) ∼ λtα, then a.s. When A (t)∼ λekt, then the flow between states must be normalized by the Laplace–Stieltjes transform of the conditional holding time distribution, in order to make the ratio an unbiased estimator of ρij.

The classical Crofton formula is generalized to the case of thick sections, and the corresponding corrective formulae are given.

Formulae relating properties of a two-phase structure to properties of a section through that structure are well known in stereology. An exact description of the error introduced into these formulae when the projection of a thick slice is observed rather than a true section (i.e. a slice of zero thickness) is presented for a model of opaque particles distributed according to a homogeneous Poisson point process in En.

A class of limit theorems involving asymptotic normality is derived for stationary processes whose spectral density has a singular behavior near frequency zero. Generally these processes have ‘long-range dependence’ but are generated from strongly mixing processes by a fractional integral or derivative transformation. Some related remarks are made about random solutions of the Burgers equation.

Let Gn (x) be the distribution of the nth successive maximum of a random walk on the real line. Under conditions typical for complete exponential convergence, the decay of Gn (x) – limn→∞ Gn(x) is asymptotically equal to H(x) γn n–3/2 as n → ∞where γ < 1 and H(x) a function solely depending on x. For the case of drift to + ∞, G∞(x) = 0 and the result is new; for drift to – ∞we give a new proof, simplifying and correcting an earlier version in [9].

A region contains n prey labeled 1, 2, …, n. Prey i is captured at the random time Zi; where Z1, Z2, …, Zn are i.i.d. with distribution function F. The statistician must decide when to stop searching, with the goal of maximizing the number of prey captured minus a linear time cost, c. The optimal strategy and its expected payoff are studied asymptotically as n, c →∞, for F a beta or Weibull distribution.

A well-known maintenance procedure for stochastically failing units is the age replacement policy (ARP). Under the ARP a unit is replaced on failure, or when it reaches a predetermined critical age.

In previous papers concerned with this procedure the authors searched for the optimal critical age to minimize the objective function under discussion. The purpose of this paper is to prove that the ARP itself is the optimal decision rule amongst all reasonable policies.

The problem of detecting the arrival of a ‘disorder' in a process observed through a monitoring facility which may operate in ‘slow’ or ‘fast’ mode, is formulated as an optimal stopping problem. It is shown that in all circumstances where there may exist an optimal policy specifying the mode of observation and the time of stopping there is a unique policy satisfying certain necessary conditions of optimality; the various circumstances and control policies are specified.

Messages arrive at a group of service channels in accordance with a time-dependent Poisson process. An arrival either (i) immediately begins k -stage Markovian service if an empty channel is reached, or (ii) balks and enters a retrial population if the channel sought is busy. Diffusion approximations to the number of messages in service (each stage) and in the retrial population are derived by writing stochastic differential equations (I + 0). Steady-state distributions are found and compared with certain simulation results.

Two independent i.i.d. sequences of random variables {Un} and {Dn} generate a Markov process {Xn} by Xn = max(Xn–1 – Dn, Un), n = 1, 2, …. ‘Exchange’ is defined as the event [Un > Xn–1 – Dn]. Conditions for existence of a limiting distribution for {Xn} are established, and normalization is discussed when no limiting distribution exists. Finally the process {Xn at the k th exchange; k = l, 2, …} and the time between consecutive exchanges are considered.

An analogue of the deterministic pandemic threshold theorem for a measles epidemic is established for a viral host-vector epidemic. The severity of the epidemic is shown to be the unique positive solution of a pair of equations.

Selective predation is an example of biased sampling which gives rise to a non-central multivariate hypergeometric distribution. The probability distribution function of this distribution is derived.

For a subcritical Bellman-Harris process for which the Malthusian parameter α exists and the mean function M(t)∼ aeat as t → ∞, a necessary and sufficient condition for e–at (1 –F(s, t)) to have a non-zero limit is known. The corresponding condition is given for the generalized branching process.

If a subcritical Galton-Watson process is initiated with an arbitrary mass distribution, then it is known that under certain conditions proper conditional limit distributions exist, depending on a single parameter. It is shown here that there is a one-to-one correspondence between these distributions and those arising from the process with a linear offspring probability generating function.

In this note, necessary and sufficient conditions for the central limit theorem for the number of events in a doubly stochastic Poisson process are given.