The multitype Galton-Watson process is considered both with and without immigration. Proofs are given for the existence of invariant measures and their uniqueness is examined by functional equation methods. Theorem 2.1 proves the uniqueness, under certain conditions, of solutions of a multidimensional Schröder equation. Regular variation is shown to play a role in the multitype theory.

We investigate the large-sample behaviour of maximum likelihood estimates (MLE's) of the parameters of a diffusion process, which is observed throughout continuous time. The results (limit normal distribution for the MLE and an asymptotic chi-squared likelihood ratio test) correspond exactly to classical asymptotic likelihood results, and follow easily from a central limit theorem for stochastic integrals.

A random collision process with transition probabilities belonging to the same type of distribution is considered. It was proved that if the characteristic function of the initial distribution has a positive radius of convergence, then the sequence {Fn(x)} converges weakly to a distribution G(x) [9]. We define a metric e1[Fn;G], which is analogous to the functional e[f] introduced by Tanaka to Kac's model of Maxwellian gas [10]. We prove that e1[Fn; G] is monotone non-increasing as n → ∞, and also the convergence of the sequence {Fn(x)} under the weaker assumption that for some a > 1 the initial distribution has an ath moment.

A discrete time, scalar, pursuit-evasion game is presented in which an evader, moving according to a stationary stochastic process, is continually being followed by a pursuer. Both players have perfect observations of the evader's positions, but the observations of the pursuer are subject to a time lag. It is assumed that the strategy of the evader can be represented as an infinite moving average, and that he is restricted by a velocity constraint. The pursuer is limited to strategies linear in his information, and the payoff is taken to be the mean-square distance between pursuer and evader. Under these conditions it is shown that the game does not have a value, and subsequently the lower and upper values and corresponding strategies are found.

The bivariate Poisson point process introduced in Cox and Lewis (1972), and there called the bivariate delayed Poisson process, is studied further; the process arises from pairs of delays on the events of a Poisson process. In particular, results are obtained for the stationary initial conditions, the joint distribution of the number of operative delays at an arbitrary time, the asynchronous counting distribution, and two semi-synchronous interval distributions. The joint delay distribution employed allows for dependence and two-sided delays, but the model retains the independence between different pairs of delays.

Let η =Σjδτj be a point process on some space S and let β,β1,β2, … be identically distributed non-negative random variables which are mutually independent and independent of η. We can then form the compound point process ξ = Σjβjδτj which is a random measure on S. The purpose of this paper is to study the limiting behaviour of ξ as . In the particular case when β takes the values 1 and 0 with probabilities p and 1 –p respectively, ξ becomes a p-thinning of η and our theorems contain some classical results by Rényi and others on the thinnings of a fixed process, as well as a characterization by Mecke of the class of subordinated Poisson processes.

Let Gn(x) be the distribution function of the maximum of the successive partial sums of independent and identically distributed random variables and G(x) its limiting distribution function. Under conditions, typical for complete exponential convergence, the decay of Gn(x) — G(x) is asymptotically equal to c.H(x)n−3/2γn as n → ∞ where c and γ are known constants and H(x) is a function solely depending on x.

Let X(t) be a continuous time Markov process on the integers such that, if σ is a time at which X makes a jump, X(σ)– X(σ–) is distributed independently of X(σ–), and has finite mean μ and variance. Let q(j) denote the residence time parameter for the state j. If tn denotes the time of the nth jump and Xn ≡ X(tb), it is easy to deduce limit theorems for from those for sums of independent identically distributed random variables. In this paper, it is shown how, for μ > 0 and for suitable q(·), these theorems can be translated into limit theorems for X(t), by using the continuous mapping theorem.

This paper considers the discrete time Markov decision model with a denumerable state space and finite action space. Under certain conditions it is proved that the minimal total expected cost for a planning horizon of n epochs minus n times the minimal long-run average expected cost per unit time has a finite limit as n → ∞ for each initial state.

Given two random k-ary sequences of length n, what is f(n,k), the expected length of their longest common subsequence? This problem arises in the study of molecular evolution. We calculate f(n,k) for all k, where n ≦ 5, and f(n,2) where n ≦ 10. We study the limiting behaviour of n–1f(n,k) and derive upper and lower bounds on these limits for all k. Finally we estimate by Monte-Carlo methods f(100,k), f(1000,2) and f(5000,2).

Discrete time extremal processes with a continuous underlying c.d.f. are random measures which can be viewed as two-dimensional Poisson processes and this representation is used to obtain the conditional law of the sequence of states the process passes through (upper record values) given the sequence of holding times in states (inter-record times). In addition the Gamma processes (which lead to the Ferguson Dirichlet processes) and a random measure that arises in sampling from a biological population are discussed as two-dimensional Poisson processes.

Let Sn, n = 1, 2, ‥, denote the partial sums of i.i.d. random variables with the common distribution function F and positive, finite mean. Let N(c) = min [k; Sk > c‥kp], c ≥ 0, 0 ≤ p < 1. Under the assumption that F belongs to the domain of attraction of a stable law with index α, 1 < α ≤ 2, functional central limit theorems for the first passage time process N(nt), 0 ≤ t ≤ 1, when n → ∞, are derived in the function space D[0,1].

Paz (1963), (1971) has shown that for an n × n stochastic matrix P the property that some power of P has a positive ergodic coefficient is decidable. In particular he shows that α (Pk) > 0 for some k, only if it is positive for k = ½n(n – 1). However he states that it is not known whether this bound is sharp. In this paper a sharp bound is given, namely k =½(n – 1)2 + ½ or ½(n − 1)2 + 1 depending on whether n is even or odd. The proof of this is based on some results from number theory.

A sufficient condition is given for a supercritical generalized age-dependent branching process, properly normalized, to converge to zero. This condition is the strongest one possible, for if it does not hold, then it is known that the normalized process does indeed converge to a legitimate random variable.

A single server is fed by a renewal stream of individual customers. These are of type k with probability πk, k = 1, …, N, and are all served individually. Upon completion of a service the server proceeds immediately with a customer of the lowest type (= highest priority) present, if any. Service times for type k are drawn from a general distribution function Bk(t) concentrated on (0, ∞).

We lay the foundations for a broad analysis of the model.

We discuss here an extension of a queueing model studied by Neuts and also by Çinlar. We obtain a matrix form of Takács' equations and exhibit the equilibrium conditions. We also show that the conditions imposed by Neuts and by Çinlar in order to obtain their results concerning the busy period are not necessary.

The provision of easily calculable measures of effectiveness for generalised Erlang queues (state-dependent parameters λn and μn of arrival and of service) motivates speculation about the applicability of renewal theory. The application envisaged is justified by known results for certain models and its extension to an operationally more promising system is proposed. Use of the formula ‘L = λW' with ‘effective’ λ calculated by foregoing methods is likewise shown to be justified by known results for certain models and hence its wider applicability is conjectured. Mechanisms are discussed which may provide improved models, and investigation is made of choices of λn and μn which may lead to time dependent solutions having a prescribed form. The example of panic buying is considered as an example.

A density model for various phenomena with densities much smaller than those explored by the Zipf model is developed. The density of spherical random packing in n dimensions serves as a model for relative frequency of length of monosyllabic words in English and also for the relative frequency of degree holders in the U.S. at various levels of educational attainment.

Given a ‘black box’ of logical elements, a sequence of independent random pulses is fed in and the output from the black box is observed. A procedure is given for estimating the parameters of the black box, in the case when the black box is constructed in a manner analogous to the classical autoregressive scheme.

We consider a fixed size Markov chain model which suffers losses and admits controlled recruitment. The family of n-step maintainable structures is described geometrically and examined.