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.

Viruses are small particles of RNA (Ribonucleic Acid) or DNA (DeoxyRibonucleic Acid) wrapped in a protein coat, which can be crystallized into a variety of regular geometric, often polyhedral, shapes. They are much smaller than bacteria, and are capable of passage through filters designed to, arrest these. Among the numerous viruses, bacteriophages (or bacterial viruses), called phages for short, have been the subject of much concentrated study. Over the past twenty years in particular, their structure, parasitic cycle, and most recently part of their genetic mapping, have been elucidated; the isolated DNA strand used by phages as their genetic information carrier, makes them eminently suitable in investigations on the molecular basis of life.

This paper applies the theory of semi-Markov processes to the construction of a stochastic model for interpreting data obtained from clinical trials. The model characterizes the patient as being in one of a finite number of states at any given time with an arbitrary probability distribution to describe the length of stay in a state. Transitions between states are assumed to be chosen according to a stationary finite Markov chain.

Other attempts have been made to develop stochastic models of clinical trials. However, these have all been essentially Markovian with constant transition probabilities which implies that the distribution of time spent during a visit to a state is exponential (or geometric for discrete Markov chains). Markov models need also to assume that the transitions in the state of a patient depend only on absolute time whereas the semi-Markov model assumes that transitions depend on time relative to a patient. Thus the models are applicable to degenerative diseases (cancer, acute leukemia), while Markov models with time dependent transition probabilities are applicable to colds and epidemic diseases. In this paper the Laplace transforms are obtained for (i) probability of being in a state at time t, (ii) probability distribution to reach absorption state and (iii) the probability distribution of the first passage times to go from initial states to transient or absorbing states, transient to transient, and transient to absorbing. The model is applied to a clinical study of acute leukemia in which patients have been treated with methotrexate and 6-mercaptopurine. The agreement between the data and the model is very good.

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.

Considered are two mutually independent recurrent processes each consisting of a time series of unitary stimuli. The durations of the intervals between the stimuli in each series are independent of each other and identically distributed with probability density functions φ (t) and ψ (t). Every stimulus of the ψ (t) process annihilates the next stimulus of the φ (t) process. The probability density function of the intervals of the transformed φ (t) process is derived for the case where either the φ (t) or the ψ (t) process is Poisson.

Several authors have recently discussed the asymptotic properties of stochastic populations which diffuse randomly throughout a given region. Sevast'yanov ([8], [9]) has investigated the extinction probability of a Markovian population in a compact region with an absorbing boundary, his analysis being in terms of “generation times”. Adke and Moyal have considered the spatial dispersion of a population which multiplies according to a simple time-dependent birth-and-death process and undergoes Gaussian diffusion on the real line ([2] and [3]) or on a finite interval with reflecting boundaries [1]. A serious limitation in Adke and Moyal's asymptotic results is that they are conditional upon a finite number of survivors. Moyal [7] has also obtained some basic formulae for a Markovian population diffusing through a general space.

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]).

First consider two particles diffusing on the same line, with positions at time t given by y1(t) and y2(t) respectively. We suppose that they cannot pass one another, so that if initially y1(0) < y2(0), then . As long as the two particles are not in contact, we suppose that each moves, independently of the other, according to the Wiener process. We must also prescribe what happens when collisions occur. There seems to be no unique natural way to do this, and we shall adopt a method recommended by its mathematical convenience. However, we shall see that there are two possible motivations for the prescription, one in terms of random walks, the other in terms of a more mechanical model.

Through the use of a zone mapping transformation, geometrical probability is applied to the problem of measuring the area of a planar figure by counting the points covered when the figure is placed at random on a regular point lattice. An expression for the variance σ2 is obtained as a function of radius for the case of a circle. This expression is equivalent to a more general formula for σ2 derived earlier by Kendall and Rankin using analytical methods. Computer calculations of σ are given for two special cases and the results are compared with those predicted by alternative asymptotic formulae also due to Kendall and Rankin. Some examples of the use of zone mapping to treat asymmetric figures, not subject to analytical methods, are described.

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 the Ehrenfest model with continuous time one considers two urns and N balls distributed in the urns. The system is said to be in state i if there are i balls in urn I, N − i balls in urn II. Events occur at random times and the time intervals T between successive events are independent random variables all with the same negative exponential distribution

When an event occurs a ball is chosen at random (each of the N balls has probability 1/N to be chosen), removed from its urn, and then placed in urn I with probability p, in urn II with probability q = 1 − p, (0 < p < 1).

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.

In this paper we present a method for obtaining explicit results for some two-sided boundary problems involving sums of independent random variables with alternating distributions. We apply the method to finding the first passage time to either one of two finite barriers, and to some situations arising in queueing and dam theory. The results can be expressed in terms of a finite sum of simple repeated integrals (or sums) of known functions (cf. formulae (3.6)– (3.11)).

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.

A particle that is located at the position (0, 0) in the (x, y) plane at the moment x = 0 is subjected to random impulses at the moments of time x = 1, 2, …, a + b. As the result of each impulse, the particle may be displaced either by a positive or by a negative step with respect to the y-axis; it is given that the particle is restricted by the condition that it has exactly a positive and b negative displacements. Hence in the (x, y) plane the path of the particle at each impulse is depicted by a shift of a unit to the right and by one unit upward or downward; that is the particle's path starts at (0, 0) and terminates at (a + b, a – b) after a + b steps. We assume that all possible different paths are equally probable, and assume also, without loss of generality, that a ≧ b.

A previous paper, hereafter referred to as “I”, reviewed the structure of a class of “regular” diffusion and birth-death processes in one dimension. Such processes were seen to have a basic simplicity manifesting itself in two ways: (1) The ergodic processes have an ergodic distribution that one may write down at once; (2) All such processes have transition functions for which a simple spectral representation is available. The simple spectral structure affords theoretical insight into the structure of related passage time densities, and numerical techniques for their evaluation. In particular, a variational principle is available for estimating the principal eigenvalues and eigenvectors.

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)).