A study is made of a series of stochastic growth processes related to multi-type branching models with interaction phenomena among the types with aim to ascertain criteria for extinction or non-extinction of the population. It is established that trends depicting changes of expected sizes of types generally overwhelm any effects of statistical fluctuations such that the conditions for extinction reduce to natural conditions on expected values. Three models are developed. The first two involve special mating systems for certain two sex populations. The last model is a neutralization phenomenon for two types of particles.

We consider immigration-branching processes constructable from an inhomogeneous Poisson process, a sequence of population probability distributions, and a homogeneous branching transition function. The set of types is arbitrary, and the process parameter is allowed to be discrete or continuous. For the branching part a weak form of positive regularity, criticality, and the existence of second moments are assumed. Varying the conditions on the immigration law, we obtain several results concerning asymptotic extinction, the rate of extinction, and limiting distribution functions of properly normalized, vector-valued counting processes associated with the immigration branching process. The proofs are based on the generating functional method.

The inner product associated with the l2 analog of the Mycielski-Ulam l1 genealogical distance serves to quantify the notion of genealogical relatedness in a fairly natural way. The expected value of the relatedness is obtained for certain random population models with stratified generations, including the Mycielski-Ulam model.

The asymptotic statistical distribution of tensile strength of a bundle of parallel fibers is determined in terms of the statistical characteristics of the individual fibers as the number of fibers in the bundle grows indefinitely large.

Let observations (X1, X2, …, Xn) be obtained from a time series {Xt} such that where the ɛt are independently and identically distributed random variables each having mean zero and finite variance, and the gu(θ) are specified functions of a vector-valued parameter θ. This paper presents a rigorous derivation of the asymptotic distributions of the estimators of A, B, ω and θ obtained by an approximate least-squares method due to Whittle (1952). It is a sequel to a previous paper (Walker (1971)) in which a similar derivation was given for the special case of independent residuals where gu(θ) = 0 for u > 0, the parameter θ thus being absent.

The intrinsic random functions (IRF) are a particular case of the Guelfand generalized processes with stationary increments. They constitute a much wider class than the stationary RF, and are used in practical applications for representing non-stationary phenomena. The most important topics are: existence of a generalized covariance (GC) for which statistical inference is possible from a unique realization; theory of the best linear intrinsic estimator (BLIE) used for contouring and estimating problems; the turning bands method for simulating IRF; and the models with polynomial GC, for which statistical inference may be performed by automatic procedures.

It is shown that the set of Markov random fields and Gibbs states with nearest neighbour potentials are the same for any finite graph. The set of Markov random fields is also shown to be the same as the equilibrium states of time-reversible birth/death processes with nearest neighbour interactions defined on the graph.

Conditions are given for the family of distributions of a stationary, discrete-time, Gaussian, vector-valued time-series with covariance structure given up to a finite number of parameters to satisfy the asymptotic differentiability conditions introduced by Le Cam (1969).

The joint distributions of the exit time and exit value of a spectrally positive process, from semi-infinite and finite intervals, are derived in the form of Fourier-Laplace transforms. Also the probability that such a process makes its first exit from a finite interval via the lower end point is obtained explicitly.

For a continuous time Markov chain the time points of transitions, belonging to a subset of the set of all transitions, are observed. Special cases include the point process generated by all transitions and doubly stochastic Poisson processes with a Markovian intensity. Equations are derived for the conditional distribution of the state of the Markov chain, given observations of the point process. This distribution may be used for prediction. For the forward recurrence time of the point process, distributions corresponding to synchronous and asynchronous sampling are also derived. The Palm distribution for the point process is specified in terms of the corresponding initial distribution for the Markov chain. In examples the point processes of arrivals and departures in a queueing system are studied. Two biological applications deal with estimation of population size and detection of epidemics.

A Markov process in discrete time with a finite state space is controlled by choosing the transition probabilities from a given convex family of distributions depending on the present state. The immediate cost is prescribed for each choice and it is required to minimise the average expected cost over an infinite future. The paper considers a special case of this general problem and provides the foundation for a general solution. The main result is that an optimal policy exists if each state of the system can be reached with positive probability from any other state by choosing a suitable policy.

An extremal-F process {Y (t); t ≧ 0} is defined as the continuous time analogue of sample sequences of maxima of i.i.d. r.v.'s distributed like F in the same way that processes with stationary independent increments (s.i.i.) are the continuous time analogue of sample sums of i.i.d. r.v.'s with an infinitely divisible distribution. Extremal-F processes are stochastically continuous Markov jump processes which traverse the interval of concentration of F. Most extremal processes of interest are broad sense equivalent to the largest positive jump of a suitable s.i.i. process and this together with known results from the theory of record values enables one to conclude that the number of jumps of Y (t) in (t1, t2] follows a Poisson distribution with parameter log t2/t1. The time transformation t→ et gives a new jump process whose jumps occur according to a homogeneous Poisson process of rate 1. This fact leads to information about the jump times and the inter-jump times. When F is an extreme value distribution the Y-process has special properties. The most important is that if F(x) = exp {—e–x} then Y(t) has an additive structure. This structure plus non parametric techniques permit a variety of conclusions about the limiting behaviour of Y(t) and its jump times.

Let F be a probability measure on the real line and G = Σ C(k)Fk∗ the probability measure subordinate to F with subordinator C restricted to the nonnegative integers. Let V(x) vary regularly of order p for x→ ∞ and either (1) V(x) F[x, ∞)→ α ≧ 0 or (2) V(x) C[x, ∞)→ γ ≧ 0. If ρ > 1 and F(–∞, 0) = 0, necessary and sufficient in order that V(x) G[x, ∞)→b, is that both (1) and (2) hold for suitable α and γ. For 0 ≦ ρ ≦ 1 the conditions are of different type. For two-sided F a different situation arises and only sufficient conditions are found. An application to renewal moments of negative order is given.

This paper is concerned with the general problem of finding an optimal transition matrix for a finite Markov chain, where the probabilities for each transition must be chosen from a given convex family of distributions. The immediate cost is determined by this choice, but it is required to minimise the average expected cost in the long run. The problem is investigated by classifying the states according to the accessibility relations between them. If an optimal policy exists, it can be found by considering the convex subsystems associated with the states at different levels in the classification scheme.