The purpose of this paper is to give some very simple applications of martingales to epidemics. The results are all connected with stopping times T (for instance the classical end of epidemic) and include the expression of the joint generating function Laplace transform of and and simple relations between moments of these three variables. (Here Xt and Yt respectively denote the numbers of susceptibles and carriers.) We also give several relations between different types of epidemics. Although this paper only deals with Downton's model, some of the methods are still valid for more general models.

In biological populations, several strategies for engaging in intraspecific contests may be present. Investigations of stability show that the mean strategy in use in the population can prove stable but that neutrally stable and gradually changing polymorphisms in strategy are to be expected when the contest affects reproductive success.

Zipf's laws are probability distributions on the positive integers which decay algebraically. Such laws have been shown empirically to describe a large class of phenomena, including frequency of words usage, populations of cities, distributions of personal incomes, and distributions of biological genera and species, to mention only a few. In this paper we present a Dirichlet–multinomial urn model for describing the above phenomena from a stochastic point of view.

We derive the Zipf's law under certain regularity conditions; some limit theorems are also obtained for the urn model under consideration.

Consider the pieces of a randomly broken stick. How long is the j th longest piece? How many breaks are necessary for getting all pieces less than a given length? These and related questions are studied in particular when the number of pieces is large. Using simple properties of the exponential distribution new proofs are given of old results and new results are obtained.

Let F be the infinite-horizon minimal cost function, and FKS the minimal cost function for horizon s and terminal cost function K. In Section 2 we define the ‘stable domain' (the set of K for which ), determine some of its properties, and relate these to the questions of stability of the process (whether ) and whether a given solution of the dynamic programming equation can be identified with F These ideas are developed in Sections 3 and 5 for various strengthenings and weakenings of the hypothesis of non-negative costs. In Section 5 we derive a new sufficient condition for stability and for characterisation of F.

In this article we examine an R & D project in which the project status changes according to a diffusion process. The decision variables include a resource expenditure strategy and a stopping policy which determines when the project should be terminated. The drift and the diffusion parameters of the project status process are assumed to be functions of the resource expenditure rate.

The terminal reward from the project is a non-decreasing function of the project status. Our purpose is to select optimal investment strategies under the discounted return criterion.

The value of the project is shown to be a solution of a second order, non-linear differential equation. Finally, we derive the optimal investment strategies for an R & D project in which the project status changes according to a non-homogeneous compound Poisson process by using diffusion approximation.

Vehicles whose lengths are independent identically distributed random variables with known distribution function are loaded onto ferries of fixed known length, each ferry departing as soon as it can no longer accommodate the next vehicle in the queue. We work out how much a vehicle of any particular length ought to pay for use of the ferry, as well as the expected number of vehicles per ferry and expected revenue per ferry in equilibrium.

In this paper it will be shown that by an appropriate choice of σ-fields, martingale methods can be used to obtain simple proofs of many of the central limit theorems known for triangular arrays of exchangeable random variables.

Consider an array of binary random variables distributed over an m1(n) by m2(n) rectangular lattice and let Y1(n) denote the number of pairs of variables d, units apart and both equal to 1. We show that if the binary variables are independent and identically distributed, then under certain conditions Y(n) = (Y1(n), · ··, Yr(n)) is asymptotically multivariate normal for n large and r finite. This result is extended to versions of a model which provide clustering (repulsion) alternatives to randomness and have clustering (repulsion) parameter values nearly equal to 0. Statistical applications of these results are discussed.

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.

It is shown that the limiting random variable W(si) of an irregular branching process with infinite mean, defined in [5], has a continuous and positive distribution on {0 < W(si) < ∞}. This implies that for all branching processes (Zn) with infinite mean there exists a function U such that the distribution of V = limnU(Zn)e–n a.s. is continuous, positive and finite on the set of non-extinction. A kind of law of large numbers for sequences of independent copies of W(si) is derived.

A variant of the Trotter–Kato theorem due to Kurtz (1969) is used to give new and simpler proofs of functional central limit theorems for Markov processes. Applications include theorems of Bellman and Harris (1951), Stone (1961), Karlin and McGregor (1965), Gihman and Skorokhod (1972) and Rosenkrantz (1975). In addition our methods yield a novel counterexample to the so-called ‘diffusion approximation'.

For a Markov chain with optional transitions, except for those to an arbitrary fixed state accessible from all others, Kesten and Spitzer proved the existence of a control policy which minimized the expected time to reach the fixed state and for constructing an optimal policy, proposed an algorithm which works in certain cases. For the algorithm to work they gave a sufficient condition which breaks down if there are countably many states and the minimal hitting time is bounded. We propose a modified algorithm which is shown to work under a weaker sufficient condition. In the bounded case with countably many states, the proposed sufficient condition is not necessary but a similar condition is. In the unbounded case as well as when the state space is finite, the proposed condition is shown to be both necessary and sufficient for the original Kesten–Spitzer algorithm to work. A new iterative algorithm which can be used in all cases is given.

A controlled Markov chain with finite state space has transition probabilities which depend on an unknown parameter α lying in a known finite set A. For each α, a stationary control law ϕ α is given. This paper develops a control scheme whereby at each stage t a parameter α t is chosen at random from among those parameters which nearly maximize the log likelihood function, and the control ut is chosen according to the control law ϕ αt. It is proved that this algorithm leads to identification of the true α under conditions weaker than any previously considered.

In this paper an axiomatic development of multistate systems is presented. Three types of coherence based on the strength of the relevancy axiom are studied. The strongest of these has been investigated previously by El-Neweihi, Proschan, and Sethuraman [3]. One of the weaker types of coherence permits wider applicability to real life situations without sacrificing any of the mathematical results obtained by El-Neweihi, Proschan, and Sethuraman. The concept of system performance is formalized through expected utility and the effect of component improvement on system performance is studied using a generalization of Birnbaum's reliability importance.

A single device shock model is studied. The device is subject to some damage process. Under the assumption that as the cumulative damage increases, the probability that any additional damage will cause failure increases, we find sufficient conditions on the shocking process so that the life distribution will be increasing failure rate.

In this paper a unified approach is used for proving relationships between customer-stationary and time-stationary characteristics of service systems with varying service rate and point processes. This approach is based on an intensity conservation principle for general stationary continuous-time processes with imbedded stationary marked point processes. It enables us to work under weaker independence assumptions than usual in queueing theory.

By means of a general intensity conservation principle for stationary processes with imbedded marked point processes (PMP) stochastic inequalities are proved between customer-stationary and time-stationary characteristics of queueing systems G/G/s/r.

The method of invariant imbedding, used extensively in astrophysics and in neutron transport theory, is applied to the analysis of dams whose input rate is a finite state-space continuous-parameter Markov chain. Passage-time distributions are considered as functions of the dam capacity. When the release rate is constant the method yields first-order non-linear differential equations with initial conditions obtained from the easily derived behaviour of a dam with zero capacity. Solutions of these equations are given. Some possible correlation functions for the increments of the cumulative input process are examined when the underlying Markov chain is stationary. Finally the analysis is extended to deal with dams having content-dependent release-rate of the form

A criterion is given for positive recurrence of a multidimensional, aperiodic, irreducible Markov chain with a denumerable state space. This criterion extends to the multidimensional case Foster's one-dimensional criterion. The multidimensional criterion consists of several conditions, one for each coordinate of the process. The usefulness of this criterion is shown through a queueing network example.