This note points out an error made in an earlier paper treating a diffusion equation for the probability distribution of gene frequencies under selection. The main interest lies in the orthogonality properties of certain eigenfunctions and the determination of the corresponding Fourier coefficients. Two particular cases of selection parameter values are considered. Approximate formulas are given for the two smallest eigenvalues for arbitrary values of the selection parameters.

The extinction probability q of a supercritical simple branching process is well known to be less than unity. Intuitively, it is apparent that when the offspring mean is close to one, so, usually, will q be. This notion is made rigorous, and simple bounds are given for q in terms of the second and third factorial moments, which are applicable when the offspring mean is close to unity. A comparison is made of various upper bounds for q. The note contains some numerical examples.

We consider the Bienaymé–Galton–Watson model of population growth in which immigration is allowed. When the mean number of offspring per individual, α, satisfies 1 < α < ∞, a well-known result proves that a normalised version of the size of the n th generation converges to a finite, positive random variable iff a certain condition is satisfied by the immigration distribution. In this paper we obtain some non-linear limit theorems when this condition is not satisfied. Results are also given for the explosive case, α = ∞.

We find the first-passage probability that X(t) remains above a level a throughout a time interval of length T given X(0) = x0 for the particular stationary Gaussian process X with mean zero and (sawtooth) covariance P(τ) = 1 – α | τ |, | τ | ≦ 1, with ρ(τ + 2) = ρ(τ), – ∞ < τ < ∞. The desired probability is explicitly found as an infinite series of integrals of a two-dimensional Gaussian density over sectors. Simpler expressions are found for the case a = 0 and also for the unconditioned probability that X(t) be non-negative throughout [0, T]. Results of some numerical calculations are given.

Using an integral equation of Darling and Siegert in conjunction with the backward Kolmogorov equation for the transition probability density function, recurrence relations are derived for the moments of the time of first exit of a temporally homogeneous Markov process from a set in the phase space. The results, which are similar to those for diffusion processes, are used to find the expectation of the time between impulses of a Stein model neuron.

The equilibrium probability distribution over the set of absorbing states of a reducible Markov chain is specified a priori and it is required to obtain the constrained sub-space or feasible region for all possible initial probability distributions over the set of transient states. This is called the inverse problem. It is shown that a feasible region exists for the choice of equilibrium distribution. Two different cases are studied: Case I, where the number of transient states exceeds that of the absorbing states and Case II, the converse. The approach is via the use of generalised inverses and numerical examples are given.

The joint probability density function of the occupation time of a three-state stochastic process with constant transition matrix: aij ≠ 0, i ≠ j, i, j = 1, 2, 3 but αii = 0, i = 1, 2, 3 is studied.

A waiting time without memory, or age-independent residual life-time, is a positive-valued random variable T with the property that for any x, y > 0, given that T >x, the conditional probability of T >x +y is the same as the unconditional probability of T >y; in other words, the physical process operates as if it has no memory concerning the successive occurrences of a certain event. The paper investigates the consequences of defining the property of lack of memory on more general time-domains than the positive reals. As a side issue, there is discussion of a stochastic variation of Cauchy's functional equation.

A basic result in the theory of total positivity is that the convolution of any two Pólya frequency sequences is again a Pólya frequency sequence. The like result for binomial convolution, associated with exponential generating functions, is proved. This and similar results are used to obtain an upper bound on the probability that the union of independent random subsets of a finite set N is all of N. Parallels from the theory of reliability involving sums of random variables with increasing failure rates are noted.

For stationary isotropic random functions on a Euclidean space, we characterize and compare the mean values of certain geometric measures of the smoothness of realizations. In particular we examine mean properties of the contours and gradients of the random function, and the effect of local averaging on smoothness in special cases.

This paper formulates the random process of line-segments in the Euclidean plane. Under conditions more general than Poisson, expressions are obtained, for Borel A ⊂ R2, for the first moments of M(A), the number of segment mid-points in A; N(A), the number of segments which intersect with convex A; S(A), the total length within A of segments crossing A; and C(A) the number of segment-segment crossings within A. In the case of Poisson mid-points, the distribution of the rth nearest line-segment to a given point is found.

Consider a system that is subject to a sequence of randomly occurring shocks; each shock causes some damage of random magnitude to the system. Any of the shocks might cause the system to fail, and the probability of such a failure is a function of the sum of the magnitudes of damage caused from all previous shocks.

The purpose of this paper is to derive the optimal replacement rule for such a system whose cumulative damage process is a semi-Markov process. This allows for both the time between shocks and the damage due to the next shock to be dependent on the present cumulative damage level.

Only policies within the class of control-limit policies will be considered; namely, policies with which no action is taken if the damage is below a fixed level, and a replacement is made if the damage is above that.

An example will be given illustrating the use of the optimal replacement rule.

This paper presents a modification to the traditional technique of approximating general queueing problems by the Erlang distributions. It presents both theory and the results of some numerical testing.

Trajstman (1974) has shown that two different population models used to study the number of mutant forms maintained in a population have a certain marginal stationary distribution in common. In this note a general stochastic population model is proposed which subsumes these two models and shows that their transition rates are also related.

In the paper the problem of the extinction of non-homogeneous Galton-Watson processes with one type of particle is studied. It is proved that the process becomes extinct if the expectations of the process do not converge to a limit (finite or infinite). If the expectations have a finite limit, then simple necessary and sufficient conditions for the extinction are proved. The general case remains open; however two more sufficient conditions which are also necessary under some restrictions are given.

It is shown that if ϕ is a given function out of a large class satisfying a certain regularity condition, then a supercritical age-dependent branching process {Z(t)} exists with deterministic immigration and given life-length and family-size distributions such that Z(t)/(eat ϕ(t)) converges in probability to a non-zero constant, a being the appropriate Malthusian parameter.

As an easy corollary one discovers the asymptotic behaviour of some processes with random immigration.

It is shown that a Galton-Watson process in varying environment, unlike a classical Galton-Watson process, can have an infinite (though at most countable) number of distinct rates of growth.

A functional central limit theorem extending the central limit theorem of Chung (1954) for the Robbins–Munro procedure is proved. It is shown that the asymptotic normality is preserved under certain random stopping rules.

Envision a one-dimensional system of infinitely many identical particles, in which initial particle positions constitute a Poisson random measure and the initial velocity of a particle depends only on its initial position. Given its initial conditions the system evolves deterministically, by means of perfectly elastic collisions. In this note we derive conditions for continuity of the probability laws of the system and of the particle paths, as functions of the parameters of the initial conditions. These results have the physical interpretation of stability theorems.

There exist three classes of probability laws that are stable for maxima. A number of well-known distributions lie in the domains of attraction of these laws. This fact is sometimes exploited by fitting the distribution of maxima with one of the stable laws. Such a procedure may well be misguided, however, since distributions exist which produce maxima having any desired distribution and not just a stable type. In this paper partial attraction of maxima is defined and it is shown that all distributions have a non-empty domain of partial attraction of maxima. In fact, there exists a distribution that lies simultaneously in the domain of partial attraction of maxima of all distributions.