This account is meant to serve as an introduction to certain mathematical ideas which can be of considerable importance in applied probability and in statistical theory. Since a wide field is to be covered in a short space, the account has to depend on heuristic arguments and a sympathetic understanding from those familiar with the technical details of the subject is called for. Constant reference to the sources of the underlying mathematics is not called for, as it will be clear that the author is not claiming these results as his own. The references at the end of the paper, which are arranged so as to aid those readers who wish to proceed further, will also indicate the sources of the mathematical and statistical ideas used. We assume some knowledge of the groups, though most concepts are defined below. By beginning from the finite dimensional case and motivating later concepts in terms of this we hope to make the later, rather technical sections, more accessible.

The present paper generalizes the results of the author's paper [2] on “Discontinuous Markov processes” to processes that are incompletely specified. These results are then used to develop a step-by-step method of solution for processes involving several kinds of “jumps”, where at the first step one solves for the transition probabilities involving jumps of the first kind, but not of the second, third, etc., at the second step for transition probabilities involving jumps of the first and second, but not the third, fourth, etc. kinds, and so on. Some examples are given.

Upper and lower bounds are given for the probability that a separable random process X(t) will take values outside the interval (— λ1, λ2) for 0 ≦ t ≦ T, where λ1 and λ2 are positive constants.

The random process needs to be neither stationary, Gaussian nor purely random (white noise).

In engineering applications, X(t) is usually a random process decaying with time at least in the long run such as the structural response to the acceleration of ground motion due to earthquake.

Numerical examples show that the present method estimates the probability between the upper and lower bounds which are sufficiently close to be useful when the random processes decay with time.

The time to absorption from the set T of transient states of a Markov chain may be sufficiently long for the probability distribution over T to settle down in some sense to a “quasi-stationary” distribution. Various analogues of the stationary distribution of an irreducible chain are suggested and compared. The reverse process of an absorbing chain is found to be relevant.

The object of this paper is to present a probabilistic model for analyzing changes through time in a binary dyadic relation on a finite set of points. The total relation on the set takes the form of an aggregate of directed binary dyadic relations between ordered pairs of points belonging to the set; equivalently, the total relation on the set can be represented by means of a digraph or an incidence matrix isomorphic with the total relation. Such a relation, changing in its structure as time proceeds, is a reasonable mathematical model, for example, for the evolution of inter-relationships of the members of a social or any other group. A group of this kind is organized for a specific activity involving some sort of “communication” from one member to the other, and may be observed at successive discrete points in time generating statistics on the evolutionary process. (For a detailed treatment, see [3].) As a matter of fact, under suitable assumptions, the model presented here has potentialities for application in those situations which can be represented mathematically in terms of a finite set of points and an all-or-none relationship between ordered pairs of these points. Some of the other examples are communication networks, ecology, animal sociology, and management sciences (see [5]).

The health of an individual during his lifetime may be considered as a stochastic process over time. He is usually born free of disease; at some time x after his birth, he becomes the victim of a mortal disease or accident, and dies at time y ≧ x. Comparative studies of chronic diseases try to evaluate the relative forces of diseases in a population, and to estimate the survival time y − x. One type of study is of the “time-oriented” type: a group of individuals is followed by a periodic survey over several years. Several papers on these problems, and comprehensive bibliographies, are contained in the volume [3].

In this paper we analyze the correlation coefficient between members of any two colonies out of an infinite ensemble of colonies. It is assumed that each colony has the same number of members, that migration takes place between the different colonies, and that there is a constant rate of mutation in each colony. An explicit formula is derived for the correlation function and the long distance form of this function is derived. It is shown that under rather weak restrictions on the pattern of migration the asymptotic form of the correlation function is characteristic of the dimension of the model.

Several models for phage reproduction within a bacterium have recently been discussed by Steinberg and Stahl (1961), Gani (1962) and Ohlsen (1963); all are effectively birth-death type processes, with the first of these considering a single type of phage mutation. A detailed description of the biological processes involved may be found in Adams (1959), Weidel (1959), Sanders (1961) and Smith (1961). In this paper we shall consider some general models extending that of Steinberg and Stahl in which mutation may take place from normal a phages to mutant A phages.

Kendall and Moran ([4], §3.41) consider the following problem, which first arose in connexion with the acoustical design of auditoria. Imagine a convex room with perfectly reflecting walls, and suppose that a particle is projected from a point inside the room. Then it will bounce around the room, and its trajectory will consist of a large number of straight segments. The average length of these segments is the mean free path of the particle.

There are several practical situations where one requires the statistical properties of the straight path length l across a specified geometrical shape. Typical examples are the length of the path of a gamma-ray to the wall of a nuclear reactor (Primak (1956)), the length of a sound ray in a room from one reflection to the next (Kosten (1960)), or the length of a straight path across a square (Horowitz (1964)).

Certain patches of ground are favorable to the formation of fair weather cumulus clouds. These are patches which reflect rather than absorb solar heat. As the air above these patches is warmed, it will rise and, if the humidity is sufficient, the cooling effect of higher elevation will cause formation of a cloud of the type known as the fair weather cumulus. This fact is used in the present paper to develop a stochastic model for the temporal evolution of the contribution to cloud cover by fair weather cumuli.

For the queueing system GI/G/1 with both waiting-line and service-line customer impatience an integral equation for the limiting waiting-time distribution function W(x) is derived and the existence of W(x) proved when the system satisfies certain conditions. Two counter examples are constructed showing that W(x) does not necessarily exist. A sufficient condition for its existence in GI/G/1 with deterministic waiting-line impatience is given. Formal solutions for M/G/l are derived for deterministic impatience and negative exponential impatience.

Suppose X1, X2, X3,… are independent and identically distributed positive nonlattice random variables with common distribution function F(x).

A well-known theorem (Moran, 1959) relating to the equilibrium condition of a semi-infinite “Moran reservoir” with stationary independent inputs and unit output, gives (a) the probability of emptiness, and (b) the generating function of the distribution of levels, in terms of the input distribution. A further theorem (Prabhu, 1958) points out that, in a finite reservoir, the ratio of the probabilities of any two comparable levels is independent of the size of the reservoir, and and is in fact the same as the corresponding ratio for the semi-infinite reservoir. In the present note the theorems are extended to deal with the case of Markovian inputs.