This paper shows how to calculate solutions to Poisson's equation for the waiting time sequence of the recurrent M/G/l queue. The solutions are used to construct martingales that permit us to study additive functionals associated with the waiting time sequence. These martingales provide asymptotic expressions, for the mean of additive functionals, that reflect dependence on the initial state of the process. In addition, we show how to explicitly calculate the scaling constants that appear in the central limit theorems for additive functionals of the waiting time sequence.

At which (random) sample size will every population element have been drawn at least m times? This special coupon collector's problem is often referred to as the Dixie cup problem. Some asymptotic properties of the Dixie cup problem with unequal sampling probabilities are described.

We prove that the quasi-invariant measures associated to a Brownian motion with negative drift X form a one-parameter family. The minimal one is a probability measure inducing the transition density of a three-dimensional Bessel process, and it is shown that it is the density of the limit distribution limt→∞Px(X A | τ > t). It is also shown that the minimal quasi-invariant measure of infinite mass induces the density of the limit distribution ) which is the law of a Bessel process with drift.

A proposal is given for estimating the home range of an animal based on sequential sightings. We assume the given sightings are independent, identically distributed random vectors X1,· ··, Xn whose common distribution has compact support. If are the polar coordinates of the sightings, then is a sup-measure and corresponds to the right endpoint of the distribution . The corresponding upper semi-continuous function l(θ) is the boundary of the home range. We give a consistent estimator for the boundary l and under the assumption that the distribution of R1 given is in the domain of attraction of an extreme value distribution with bounded support, we are able to give an approximate confidence region.

We prove a functional law of large numbers and a functional central limit theorem for a controlled renewal process, that is, a point process which differs from an ordinary renewal process in that the ith interarrival time is scaled by a function of the number of previous i arrivals. The functional law of large numbers expresses the convergence of a sequence of suitably scaled controlled renewal processes to the solution of an ordinary differential equation. Likewise, the functional central limit theorem establishes that the error in the law of large numbers converges weakly to the solution of a stochastic differential equation. Our proofs are based on martingale and time-change arguments.

This paper is concerned with a model for the spread of an epidemic in a closed, homogeneously mixing population in which new infections occur at rate f(x, y) and removals occur at rate g(x, y), where x and y are the numbers of susceptible and infective individuals, respectively, and f and g are arbitrary but specified positive real-valued functions. Sequences of such epidemics, indexed by the initial number of susceptibles n, are considered and conditions are derived under which the epidemic processes converge almost surely to a birth and death process as n tends to infinity. Thus a threshold theorem for such an epidemic model is obtained. The results are extended to models which incorporate immigration and emigration of susceptibles. The theory is illustrated by several examples of models taken from the epidemic literature. Generalizations to multipopulation epidemics are discussed briefly.

By imbedding the multitype version of the standard epidemic model in a multiparameter process, we derive a functional limit theorem for the total cost of the epidemics.

In the theory of autoregressive model fitting it is of interest to know the asymptotic behaviour, for large sample size, of the coefficients fitted. A significant role is played in this connection by the moments of the norms of the inverse sample covariance matrices. We establish uniform boundedness results for these, first under generally weak conditions and then for the special case of (infinite order) processes. These in turn imply corresponding ergodic theorems for the matrices in question.

Consider a number of events in a probability space. Let X be a random variable that is the number of events that occur. Given some of the moments of the distribution of X, it is possible to obtain bounds on the probability that at least one event occurs. The best possible bounds are given here. If there are many equiprobable events that are d- wise independent, and d is even, then the probability that at least one event happens is at least 1 — O(µ–d/2), where μ = E(X).

Let Y1, Y2, · ·· be a stochastic process and M a positive real number. Define TM = inf{n | Yn > M} (TM = + ∞ if for n = 1, 2, ···)· We are interested in the probabilities P(TM <∞) and in particular in the case when these tend to zero exponentially fast when M tends to infinity. The techniques of large deviations theory are used to obtain conditions for this and to find out the rate of convergence. The main hypotheses required are given in terms of the generating functions associated with the process (Yn).

Let F be the gamma distribution function with parameters a > 0 and α > 0 and let Gs be the negative binomial distribution function with parameters α and a/s, s > 0. By combining both probabilistic and approximation-theoretic methods, we obtain sharp upper and lower bounds for . In particular, we show that the exact order of uniform convergence is s–p, where p = min(1, α). Various kinds of applications concerning charged multiplicity distributions, the Yule birth process and Bernstein-type operators are also given.

An exchangeable sequence of random variables is constructed with all finite-dimensional distribution functions having an Archimedean copula (as defined by Schweizer and Sklar (1983)). Through a monotone transformation of this exchangeable sequence, we obtain and characterize the class of exchangeable sequences possessing the max-stable property as defined by De Haan and Rachev (1989). Several parametric examples are given.

A central limit theorem for cumulative processes was first derived by Smith (1955). No remainder term was given. We use a different approach to obtain such a term here. The rate of convergence is the same as that in the central limit theorems for sequences of independent random variables.

Through the study of a simple embedded martingale we obtain an extension of the Kesten–Stigum theorem and prove a central limit theorem for controlled Galton-Watson processes.

We show that the one-dimensional self-organizing Kohonen algorithm (with zero or two neighbours and constant step ε) is a Doeblin recurrent Markov chain provided that the stimuli distribution μ is lower bounded by the Lebesgue measure on some open set. Some properties of the invariant probability measure vε (support, absolute continuity, etc.) are established as well as its asymptotic behaviour as ε ↓ 0 and its robustness with respect to μ.

Some exact and asymptotic joint distributions are given for certain random variables defined on the excursions of a simple symmetric random walk. We derive appropriate recursion formulas and apply them to get certain expressions for the joint generating or characteristic functions of the random variables.

We prove a generalization of Sanov's theorem in which the state space S is arbitrary and the set of probability measures on S is endowed with the τ -topology.

The accuracy of the Poisson approximation to the distribution of the numbers of large and small m-spacings, when n points are placed at random on the circle, was analysed using the Stein–Chen method in Barbour et al. (1992b). The Poisson approximation for m≧2 was found not to be as good as for 1-spacings. In this paper, rates of approximation of these distributions to suitable compound Poisson distributions are worked out, using the CP–Stein–Chen method and an appropriate coupling argument. The rates are better than for Poisson approximation for m≧2, and are of order O((log n)2/n) for large m-spacings and of order O(1/n) for small m-spacings, for any fixed m≧2, if the expected number of spacings is held constant as n → ∞.

A measure-valued diffusion approximation to a two-level branching structure was introduced in Dawson and Hochberg (1991) where it was shown that conditioned on non-extinction at time t, and appropriately rescaled, the process converges as t → ∞to a non-trivial limiting distribution. Here we discuss a different approach to conditioning on non-extinction (popular in one-level branching) and relate the two limiting distributions.

The tail behaviour of the limit of the normalized population size in the simple supercritical branching process, W, is studied. Most of the results concern those cases when a tail of the distribution function of W decays exponentially quickly. In essence, knowledge of the behaviour of transforms can be combined with some ‘large-deviation' theory to get detailed information on the oscillation of the distribution function of W near zero or at infinity. In particular we show how an old result of Harris (1948) on the asymptotics of the moment-generating function of W translates to tail behaviour.