Algorithms for a stochastic population process, based on assumptions underlying general age-dependent branching processes in discrete time with time inhomogeneous laws of evolution, are developed through the use of a new representation of basic random functions involving birth cohorts and random sums of random variables. New algorithms provide a capability for computing the mean age structure of the process as well as variances and covariances, measuring variation about means. Four exploratory population projections, testing the implications of the algorithms for the case of time-homogeneous laws of evolution, are presented. Formulas extending mean and variance functions for unit population projections to an arbitrary initial population size are also presented. These formulas show that, in population processes with non-random laws of evolution, stochastic fluctuations about the mean function are negligible when initial population size is large. Further extensions of these formulas to the case of randomized laws of evolution suggest that stochastic fluctuations about the mean function can be significant even for large initial populations.

Based on infinite and finite line of descent of particles of a one-dimensional supercritical Galton-Watson branching process (GWBP), we construct an associated bivariate process. We show that this bivariate process is a two-dimensional, supercritical GWBP. We also show that this process retains its branching property on appropriate probability spaces, when conditioned on set of non-extinction and set of extinction. Some asymptotic and weak convergence results for this process have been established. A generalisation of these results to a multitype p-dimensional GWBP has also been carried out.

A generalization of a birth and death chain in a random environment (Yn, Zn) is developed allowing for feedback to the environmental process (Yn). The resulting process is then known as a birth and death chain in a random environment with feedback. Sufficient conditions are found under which the (Zn) process goes extinct almost surely or has strictly positive probability of non-extinction.

In this paper we consider an economic agent (the consumer) having initial wealth x(0), which he is allowed to spend during the time interval [0, T]. There are N consumer goods, the prices of which are described by a system of stochastic differential equations, and the consumer can spend his money as he pleases provided that he does not get into debt. The consumer's problem is to maximize his expected utility over [0, T]. Given a separable utility function we show that the optimal consumption policy is linear in the wealth variable regardless of the structure of the price system, and if we assume that the price processes are geometric Brownian motions we can give the optimal consumption policy explicitly.

Two classes of finite and infinite moving-average sequences of bivariate random vectors are considered. The first class has bivariate exponential marginals while the second class has bivariate geometric marginals. The theory of positive dependence is used to show that in various cases the two classes consist of associated random variables. Association is then applied to establish moment inequalities and to obtain approximations to some joint probabilities of the bivariate processes.

Extreme-value theory is considered in the context of independent but not identically distributed random variables. The departure from the case of identical distributions comes from a trend added to i.i.d. observations. Unlike previous authors we consider trends that do not completely destroy the asymptotic behaviour from the i.i.d. case.

The class of stopping rules for a sequence of i.i.d. random variables with partially known distribution is restricted by requiring invariance with respect to certain transformations. Invariant stopping rules have an intuitive appeal when the optimal stopping problem is invariant with respect to the actual gain function. Uniformly best invariant stopping rules are derived for the gamma distribution with known shape parameter and unknown scale parameter, for the uniform distribution with both endpoints unknown, and for the normal distribution with unknown mean and variance. Some comparisons with previously published results are made.

A transient renewal process based on a sequence of possibly infinite waiting times is defined. The process is studied when the (rescaled) distribution of the waiting times belongs to the subexponential class of distributions. In this case, even conditional on all waiting times observed by time t being finite, the distributions of the forward and backward delays at t are asymptotically degenerate. Also, the conditional moments of the number of events by time t converge to the same finite limits as the unconditional moments.

Stochastic optimization of semimartingales which permit a dynamic description, like a stochastic differential equation, leads normally to dynamic programming procedures. The resulting Bellman equation is often of a very genera! nature, and analytically hard to solve. The models in the present paper are formulated in terms of the relative change, and the optimality criterion is to maximize the expected rate of growth. We show how this can be done in a simple way, where we avoid using the Bellman equation. An application is indicated.

Random processes of the diffusion type have the property that microscopic fluctuations of the trajectory make possible the identification of certain statistical parameters from one continuous observation. The paper deals with the construction of parameter estimates when observations are made at discrete but very dense time points.

In this paper various notions of positive and negative dependence for bivariate stochastic processes are introduced and their interrelationship is studied. Examples are given to illustrate these concepts.

We provide sufficient conditions under which two random vectors could be stochastically compared using the standard construction. These conditions are weaker than those discussed by Arjas and Lehtonen (1978) and Veinott (1965). Using these conditions we present extensions of (i) a result of Block et al. (1984) concerning the stochastic monotonicity of independent and identically distributed random variables conditioned on their partial order statistics, and (ii) a theorem of Efron (1965) regarding an increasing property of Pólya frequency functions. Applications of these extensions are also pointed out.

A composite material is a parallel arrangement of stiff brittle fibers in a flexible matrix. Under load fibers fail, and the loads of failed fibers are locally redistributed onto nearby survivors through the matrix. In this paper we develop a new technique for computing the probability of failure under a previously studied model of the failure process. A recursion and limit theorem are obtained which apply separately to static strength and fatigue lifetime depending on the composite loading and the probability model for the failure of individual fibers under their own loads. The limit theorem yields an approximation for the distribution function for composite lifetime which is of the form 1 – [1 – W(t)]mn where W(t) is a characteristic distribution function and mn is the composite volume, reflecting a size effect. A similar result holds also for static strength. In both cases such a result was conjectured several years ago. This limit theorem is obtained from the recursion upon applying a key theorem in the theory of the renewal equation. In the proofs three technical conditions arise which must be verified in specific applications. In the case of static strength these conditions are quite easy to verify, but in the case of fatigue lifetime the verification is generally difficult, and entails considerable numerical computation.

There are a number of ad hoc regression models for the statistical analysis of lifetime data, but only a few examples exist in which physical considerations are used to characterize the model. In the present paper a complete characterization of a regression model is given by solving a functional equation recurring in the literature for the case of a fatigue problem. The result is that, if the lifetime for given values of the regressor variable and the regressor variable for a given lifetime are both Weibull variables (assumptions which are well founded, at least as approximations, from extreme-value theory in some concrete applications), there are only three families of (conditional) distribution for the lifetime (or for the regressor variable). This model is then applied to a practical problem for illustration.

A model incorporating the effect of a common environment on several components (structurally independent) of a system is developed. A multivariate generalization of the Lomax (Pareto type 2) distribution is obtained by mixing exponential variables. Its relationship to other multivariate distributions is discussed. Several properties of this distribution are reported and their usefulness in reliability theory indicated. Finally, a further generalization of this multivariate Lomax distribution is presented.

Some consequences of a modified repair system for Phillips' (1981a, b) model for a two-component system are discussed. In the original model, both components are repaired whenever a revealed fault occurs; in the modified model only faulty components are repaired. Specifically (i) the distribution of time from the initial state up to discovery of an unrevealed fault, (ii) the expected proportion of time during which there exists an unrepaired fault, and (iii) the distribution of number of revealed faults up to and including the one which leads to a discovery of an unrevealed fault, are obtained. The theory is illustrated by examples, based on specific distributions for the times between repairs and occurrences of the two types of faults. A characterization of the exponential distribution is indicated.

The problem of optimal control of a finite dam in the class of policies has been considered by Lam Yeh [6], [7]. In this paper, by using the first Dynkin formula, the same problems of specifying an optimal policy in the class of the policies to minimize the expected total discounted cost as well as the long-run average cost are considered. Both the expected total discounted cost and long-run average cost are determined explicitly, and then the optimal policy can be found numerically, Also, we obtain the transition density function and the resolvent operator of a reflecting Wiener process.

For a sequence ξ1, ξ2, · ·· of i.i.d. random variables let X0 = 0 and Xk = max(Xk–1 + ξ k, 0) for k = 1, 2, ···. Let . These stopping times are used in Page's (1954) one-sided cusum procedures and are also important in queueing theory. Various asymptotic properties of Nx are derived.

We consider GI/G/1 queues in an environment which is periodic in the sense that the service time of the nth customer and the next interarrival time depend on the phase θ n at the arrival instant. Assuming Harris ergodicity of {θ n} and a suitable condition on the traffic intensity, various Markov chains related to the queue are then again Harris ergodic and provide limit results for the standard customer- and time-dependent processes such as waiting times and queue lengths. As part of the analysis, a result of Nummelin (1979) concerning Lindley processes on a Markov chain is reconsidered.