The method developed for the treatment of the classical drift models of Wright and Moran, and their generalizations, in Cannings (1974) are extended to more complex haploid models. The possibility of subdivision of the population, as for migration models and age-structured models, is incorporated. Models with variable size or reproductive structure determined by another Markov chain are analysed.

Let X be an age-dependent branching process with lifetime distribution G and age-dependent generating function π(y,s) = σk = 0∞pk(y) sk. We assume that G is right-continuous and G(0+) = G(0) = 0. The base state space S is [0,T) where T = inf{t : G(t) = 1}. Set m(y) = σk = 0∞k pk(y) and Then extinction occurs with probability one iff m ≤ 1. In the case where m > 1, define the Malthusian parameter λ to be the unique (positive) root of and set on S. is a -space-time harmonic function of the process X and the corresponding non-negative martingale converges w.p.l to a random variable W; furthermore, under a regularity assumption, W is non-trivial iff where and If 0 < a ≤ Φ ≤ β < ∞, for some constants a, β, then w.p.l, where Zt is the number of particles at time t.

This paper considers certain stochastic control problems in which control affects the criterion through the process trajectory. Special analytical methods are developed to solve such problems for certain dynamical systems forced by white noise. It is found that some control problems hitherto approachable only through laborious numerical treatment of the non-linear Bellman-Hamilton-Jacobi partial differential equation can now be solved.

We consider a non-stationary Bayesian dynamic decision model with general state, action and parameter spaces. It is shown that this model can be reduced to a non-Markovian (resp. Markovian) decision model with completely known transition probabilities. Under rather weak convergence assumptions on the expected total rewards some general results are presented concerning the restriction on deterministic generalized Markov policies, the criteria of optimality and the existence of Bayes policies. These facts are based on the above transformations and on results of Hindererand Schäl.

A right-shift process is a Markov process with multidimensional finite state space on which the infinitesimal transition movement is a shifting of one unit from one coordinate to some other to its right. A multidimensional right-shift process consists of v ≧ 1 concurrent and dependent right-shift processes. In this paper applications of multidimensional right-shift processes to some well-known examples from epidemic theory, queueing theory and the Beetle probblem due to Lucien LeCam are discussed. A transformation which orders the Kolmogorov forward equations into a triangular array is provided and some computational procedures for solving the resulting system of equations are presented. One of these procedures is concerned with the problem of evaluating a given transition probability function rather than obtaining the solution to the complete system of forward equations. This particular procedure is applied to the problem of estimating the parameters of a multidimensional right-shift process which is observed at only a finite number of fixed timepoints.

This paper surveys recent results on the covariance structure of processes generated by queues and related stochastic processes. The generated processes include the number X(t) of customers at time t, the waiting time Wn of the nth customer and the inter-departure interval Dn, i.e., the interval between departures of the nth and (n+1)th customers. The importance of these results is discussed, particularly in the field of estimation.

For the Ek/G/1 queue with finite waiting room the phase technique is used to analyse the Markov chain imbedded in the queueing process at successive instants at which customers complete service, and the distribution of the busy period, together with the number of customers who arrive, and the number of customers served, during that period, is obtained. The limit as the size of the waiting room becomes infinite is found.

For dependent probability systems of m events partially specified only by the quantities S1, the sum of the probabilities of the m individual events; S2, the sum of the probabilities of each of the (m) pairs of events and S3 the sum of the probabilities of each of the (m 3) combinations of three events; this paper develops the most stringent upper and lower bounds on P1, the probability of the union of the m events; and on P[m], the probability of the simultaneous occurrence of the m events.