The scan statistic is defined as the supremum of a particular continuous-time stochastic process, and is used as a test statistic for testing uniformity against a simple clustering type of alternative. Its distribution under the null hypothesis is investigated and weak convergence of the stochastic process to the appropriate Gaussian process is proved. An interesting link is forged between the circular scan statistic and Kuiper's statistic, which rids us of the trouble of estimating a nuisance parameter. Distributions under the alternative are then derived, and asymptotic power comparisons are made.

In 1952 Pollaczek discovered a remarkable formula for the Laplace-Stieltjes transforms of the distributions of the ordered partial sums for a sequence of independent and identically distributed real random variables. In this paper Pollaczek's result is proved in a simple way and is extended for a semi-Markov sequence of real random variables.

The maxima of independent Weiner processes spatially normalized with time scales compressed is considered and it is shown that a weak limit process exists. This limit process is stationary, and its one-dimensional distributions are of standard extreme-value type. The method of proof involves showing convergence of related point processes to a limit Poisson point process. The method is extended to handle the maxima of independent Ornstein–Uhlenbeck processes.

The points of a homogeneous Poisson process within a compact convex set are observed. We consider how to reconstruct this domain from the observations.

In this paper we find the distribution of the area X of a randomly chosen triangle within a given triangle T. An extension of the classical Crofton technique is used in setting up a sequence of differential equations to obtain the desired distribution.

A limiting distribution for the age of a class of Markov processes is found if the present state of the process is known. We use this distribution to find the age of branching processes. Using the fact that the moments of the age of birth and death processes and of diffusion processes satisfy difference equations and differential equations respectively, we find simple formulas for these moments. For the Wright–Fisher genetic model we find the probability that a given allele is the oldest in the population if all the gene frequencies are known. The proofs of the main results are based on methods from renewal theory.

We consider an infinite Markov chain with states E0, E1, …, such that E1, E2, … is not closed, and for i ≧ 1 movement to the right is limited by one step. Simple algebraic characterizations are given for persistency of all states, and, if E0 is absorbing, simple expressions are given for the probabilities of staying forever among the transient states. Examples are furnished, and simple necessary conditions and sufficient conditions for the above characterizations are given.

The problem of tendency to consensus in an information-exchanging operation is connected with the ergodicity problem for backwards products of stochastic matrices. For such products, weak and strong ergodicity, defined analogously to these concepts for forward products of inhomogeneous Markov chain theory, are shown (in contrast to that theory) to be equivalent. Conditions for ergodicity are derived and their relation to the consensus problem is considered.

In a single-shelf library having infinitely many books B1, B2, …, the probability of selecting each book is assumed known. Books are removed one at a time and replaced in position k prior to the next removal. Books are moved either to the right or the left as is necessary to vacate position k. Those arrangements of books where after some finite position all the books are in natural order (book i occupies position i) are considered as states in an infinite Markov chain. When k > 1, we show that the chain can never be positive recurrent. When k = 1, we find the limits of ratios of one-step transition probabilities; and when k = 1 and the chain is transient, we find the Martin exit boundary.

A sequence of finite point processes {Pn} is constructed in using a Markov sequence of points. Essentially, in the process Pn consisting of n events, the coordinates of these events are simply the first n points of a Markov sequence suitably scaled so that the average density of the process is independent of n. The second-order properties of Pn are discussed and sufficient conditions are found for Pn to converge in distribution to a Poisson process as n →∞. A simple example involving the cardioid distribution is described.

A construction is given for a stationary sequence of random variables {Xi} which have exponential marginal distributions and are random linear combinations of order one of an i.i.d. exponential sequence {εi}. The joint and trivariate exponential distributions of Xi−1, Xi and Xi+ 1 are studied, as well as the intensity function, point spectrum and variance time curve for the point process which has the {Xi} sequence for successive times between events. Initial conditions to make the point process count stationary are given, and extensions to higher-order moving averages and Gamma point processes are discussed.

Some relationships are derived between the asynchronous and partially synchronous counting and interval processes associated with a multivariate stationary point process. A few examples are given to illustrate some of these relationships.

The paper examines the renewal function associated with a sequence of probability distributions, which is defined by concave recurrence relations or by an even more general procedure. The elementary renewal theorem is generalized to such sequences. The results can be used to establish renewal theorems for first death in branching processes, if only the possibly generation dependent probability generating functions converge to a limit.

The second-moment structure of an estimator V*(t) of the variance-time curve V(t) of a weakly stationary point process is obtained in the case where the process is Poisson. This result is used to establish the weak convergence of a class of estimators of the form Tβ(V*(tTα) – V(tTα)), 0 < α < 1, to a non-stationary Gaussian process. Similar results are shown to hold when α = 0 and in the case where V(tTα) is replaced by a suitable estimator.

Let {Sn, n ≧ 1} be a zero, mean square integrable martingale for which so that Sn → S∞ a.s., say, by the martingale convergence theorem. The paper is principally concerned with obtaining central limit and iterated logarithm results for Bn(Sn – S∞) where the multipliers Bn ↑ ∞ a.s. An example on the Pólya urn scheme is given to illustrate the results.

Necessary and sufficient conditions are given for a distribution function in ℝ2 to be max-infinitely divisible. The d.f. F is max i.d. if Ft is a d.f. for every t > 0. This property is essential in defining multivariate extremal processes and arises in an approach to the study of the range of an i.i.d. sample.

Two examples for marked point processes are discussed and some characteristic parameters of these models are calculated. Both examples are in some way modifications of the counter models which are well known and treated in several textbooks.

In this article we present a limiting result for the random variable Yn(r) which arises in a clustering model of Strauss (1975). The result is that under some sparseness-of-points conditions the process {Yn(r): 0 ≦ r ≦ r∞} converges weakly to a non-homogeneous Poisson process {Y(r): 0 ≦ r ≦ r∞} when n → ∞. Simulation results are given to indicate the accuracy of the approximation when n is moderate and applications of the limiting result to tests for clustering are discussed.

We consider maximum likelihood estimators for parameters of diffusion processes that are generated by nth-order Ito equations. We establish asymptotic consistency as well as convergence in distribution to normality for the estimators. Examples are presented and discussed.

A model is considered for a situation in which one rumour suppresses another in a closed population. The distribution of the maximum value attained by the proportion spreading the weaker rumour is obtained in the asymptotic case, and this is compared with some actual distributions for finite population size. Closer approximations to the latter distributions are obtained.