A model is constructed to describe the evolution of a monoecious haploid population in which the population of adults remains of constant size, but for which generations may be overlapping, family size distributions are to a considerable degree arbitrary, and differential survival rates may apply with respect to adult and young individuals. The eigenvalues of the transition probability matrix are obtained, and those which determine the rate of evolution are found as simple functions of mutation rates, the mean lifetime of an individual, and the variance of his viable offspring number. The model builds on, and includes as special cases, certain non-overlapping generation models of Wright (1931), Karlin and McGregor (1965), and an overlapping generation model of Moran (1958).

One of the secondary conclusions of the study is that for some models, when evolutionary rates are measured on a mean-lifetime scale, the more offspring that are produced at a mating season may slow down evolution when mutation is absent, but may speed it up if mutation is present.

Consider an age-dependent branching process with two types of individuals. Suppose that individuals of one type beget children of both types, whereas those of the other type can only give birth to individuals of their own kind. This paper is a study of the relation between two random functions occurring in such processes starting from an ancestor of the first type, the two functions being the numbers of individuals of the two kinds. Under weak assumptions it is shown that the random proportion of individuals of one type converges as time passes, in mean square as well as almost surely to a non-random limit, easily determined in terms of the reproduction laws and life-length distributions of the process.

Although his work on the application of invariance concepts to the sequential testing of composite hypotheses is better known, Cox (1963) has also outlined a large sample approach to the same problem. His method is based on Bartlett's (1946) recognition that the sequence of maximum likelihood estimates (MLE) of the parameter of interest, calculated from an increasing number of observations, resembles asymptotically a random walk of normally distributed variables. However, the large sample theory needed to justify this approach rigorously is left largely implicit. At the end of his paper, Cox suggests that these methods may be extended to yield a sequential comparison of survival curves (Armitage (1959)), a suggestion which has been reiterated as a research problem in the monograph of Wetherill (1966).

Viral DNA is studied by the technique of radioactive labelling and subsequent ultra centrifugation which separates it from reference DNA according to length or density. One hypothesis that frequently is of service is that the strands of DNA being studied are randomly cut along their lengths. Often the average number of cuts, λ, per unit length is either known by the nature of the experiment or is desired as a statistic. If radioactive phosphorous, P32, is used to label the DNA, e.g., 6 P32 atoms per unit length of 200,000 nucleotides (each of which is connected to its neighbor by two phosphorous-containing links) then within two weeks about 3 P32 atoms will have decayed and the DNA obtained, once separated into single stranded form (denatured), will form a predictable distribution of sizes. Of course 3 decays may be obtained more quickly by starting with more than 6 P32 atoms per unit length.

Collective risk theory is concerned with the random fluctations of the total assets, the risk reserve, of an insurance company. Consider a company which only writes ordinary insurance policies such as accident, disability, fire, health, and whole life. The policyholders pay premiums regularly and at certain random times make claims to the company. A policyholder's premium, the gross risk premium, is a positive amount composed of two components. The net risk premium is the component calculated to cover the payments of claims on the average, while the security risk premium, or safety loading, is the component which protects the company from large deviations of claims from the average and also allows an accumulation of capital. When a claim occurs the company pays the policyholder a positive amount called the positive risk sum.

In [5] Renyi considers the following traffic model: Vehicles enter a highway at times 〈Ti, i = 1, 2, … 〉, forming a homogeneous Poisson process of intensity λ. The vehicle entering at time Ti will choose a velocity Vi and will travel at that constant velocity. The random variables 〈Vi, i = 1, 2, …〉 are independently and identically distributed (i.i.d.) and independent of 〈Ti〉 with c.d.f. F satisfying All vehicles travel in the same direction.

A stochastic process in which any tree in a forest planted on an integer lattice eliminates its four neighbours when it exceeds their heights, is studied for the case when all heights are independent samples from a continuous distribution. The proportion of the trees of the forest eliminated in this manner is determined for both the one and the two dimensional integer lattices.

The block replacement policy, wherein items are replaced at regular intervals of time and on failure, is rather wasteful because sometimes almost new items are also removed. As an alternative a policy of replacement by new items at regular intervals of time and by used items on failure, is suggested. The consequences of this policy, called used item replacement policy, are studied for Erlangian and sub-exponential life-time distributions. The latter distribution which is the difference of two negative exponential distributions, does not seem to have received much attention in the literature so far.

A new two parameter family of life length distributions is presented which is derived from a model for fatigue. This derivation follows from considerations of renewal theory for the number of cycles needed to force a fatigue crack extension to exceed a critical value. Some closure properties of this family are given and some comparisons made with other families such as the lognormal which have been previously used in fatigue studies.

The estimation problem is studied for a new two-parameter family of life length distributions which has been previously derived from a model of fatigue crack growth. Maximum likelihood estimates of both parameters are obtained and iterative computing procedures are given and examined. A simple estimate of the median life is exhibited, shown to be consistent and then compared, favorably, with the maximum likelihood estimate. More important, the asymptotic distribution of this estimate is shown to be within the same class of distributions as the observations themselves. This model, and these estimation procedures, are tried by fitting this distribution to several extensive sets of fatigue data and then some comparisons of practical significance are made.

A Markovian process with rewards is considered. The interference into its development consists in instantaneous changes of state called replacements. It is shown that, for any replacement policy not depending on the future, the expected reward is a linear function of the expected replacement numbers. The application of this result is illustrated by examples.

Some previously obtained asymptotic results for branching Poisson processes are extended and sharpened. It is shown that under rather general conditions the number of events in both the transient and the equilibrium processes, suitably normalized, have a unit normal distribution. Finally, unique initial conditions are derived for the equilibrium process.

In a previous paper (Leslie (1967)) the distribution of recurrence times for a particular set of success-failure patterns on a sequence of Bernoulli trials was investigated. We now consider the analogous events in continuous time and obtain the Laplace Transform (L.T.) of the distribution of recurrence times; numerical inversion yields the distribution functions.

This paper considers aspects of the following problem. Let F(x, θ) be a distribution function, d.f., in x for all θ and a Borel measurable function of θ. Define the mixture (Robbins (1948)), where Φ is a d.f., then it is of interest to determine conditions under which F(x) and F(x, θ) uniquely determine Φ. If there is only one Φ satisfying (1), F is said to be an identifiable mixture. Usually a consistency assumption is used whereby it is presumed that there exists at least one solution to (1).

1. This paper was motivated by some questions of Barnett and Lewis (1967) concerning extreme winter temperatures. The temperature during the winter can be hopefully regarded as generated by a stationary Gaussian process superimposed on a locally U-shaped trend. One is interested in statistical properties of the minimum of sample paths from such a process, and of their excursions below a given level. Equivalently one can consider paths from a stationary process crossing a curved boundary of the same form. Problems of this type are discussed by Cramer and Leadbetter (1967), extensively in the trend-free case and in less detail when a trend is present, following the method initiated by Rice (1945). While results on moments are easy to obtain, explicit results for the actual probability distributions are not usually available. However, in the important case when the level of values of interest is far below the mean, the asymptotic independence of up-crossing times makes it possible to derive simple approximate distributions. (See Cramer and Leadbetter (1967) page 256, Keilson (1966).)

Let X(t) be a stochastic process whose parameter t runs over a finite or infinite n terval T. Let t1, t2 ɛ T, t1 〈 t2; the random variable X(t2) – X(t1) is called the increment of the process X(t) over the interval [t1, t2]. A process X(t) is said to be homogeneous if the distribution function of the increment X(t + τ) — X(t) depends only on the length τ of the interval but is independent of the endpoint t. Two intervals are said to be non-overlapping if they have no interior point in common. A process X(t) is called a process with independent increments if the increments over non-overlapping intervals are stochastically independent. A process X(t) is said to be continuous at the point t if plimτ→0 [X(t + τ) — X(t)] = 0, that is if for any ε > 0, limτ→0P(| X(t + τ) — X(t) | > ε) = 0. A process is continuous in an interval [A, B] if it is continuous in every point of [A, B].

Let Xi, i = 1, 2, 3, … be a sequence of independent and identically distributed random variables which belong to the domain of attraction of a stable law of index a. Write S0= 0, Sn = Σ i=1nXi, n ≧ 1, and Mn = max0 ≦ k ≦ nSk. In the case where the Xi are such that Σ1∞n−1Pr(Sn > 0) < ∞, we have limn→∞Mn = M which is finite with probability one, while in the case where Σ1∞n−1Pr(Sn < 0) < ∞, a limit theorem for Mn has been obtained by Heyde [9]. The techniques used in [9], however, break down in the case Σ1∞n−1Pr(Sn < 0) < ∞, Σ1∞n−1Pr(Sn > 0) < ∞ (the case of oscillation of the random walk generated by the Sn) and the only results available deal with the case α = 2 (Erdos and Kac [5]) and the case where the Xi themselves have a symmetric stable distribution (Darling [4]). In this paper we obtain a general limit theorem for Mn in the case of oscillation.

Several mechanisms under which the randomness of straight line paths through convex bodies can arise are described. Some general results are given relating four of these mechanisms, and the corresponding distributions of the lengths of the straight line paths are found for the circle, the rectangle, and the cube.

Suppose Q1⋆, … Qn⋆ are independent, identically distributed random variables, each with probability density function f(x), cumulative distribution function F(x), where F(1) – F(0) = 1, f(x) is continuous in the open interval (0, 1) and continuous on the right at x = 0 and on the left at x = 1, and there exists a positive C such that f(x) > C for all x in (0, l). f(0) is defined as f(0+), f(1) is defined as f(1–).

It is well known that any stochastically continuous real valued stochastic process with independent increments defined on a compact time interval can be decomposed into a sum of independent processes, one of which is Gaussian with continuous sample paths, and the remainder of which have sample paths which are continuous except at a finite number of points with the discontinuities occurring at Poisson time points. The purpose of this note is to announce a proof of the above theorem in the case where the process takes values in an abelian group G. The detailed proof will appear elsewhere. The basic ideas of the proof in the case when G is finite dimensional Euclidean space are contained in Chapter VI of Gikhman and Skorohod (1965).