Randomly generated subsets of a point-set A0 in the k-dimensional Euclidean space Rk are investigated. Under suitable restrictions the probability is determined that a randomly located set which hits A0. is a subset of A0. Some results on the expected value of the measure and the surface area of the common intersection-set formed by n randomly located objects and A0 are generalized and derived for arbitrary dimension k.

We consider a population distributed over two habitats as represented by two separate one-dimensional branching processes with random environments. The presence of random fluctuation in reproduction rates in both habitats implies the possibility that neither habitat is universally superior to the other for all times and that a maximal population size is to be achieved by having population members present in both habitats. We show that optimal population growth occurs when migration between habitats occurs at a fixed rate which can be found from the environmentally determined reproduction variables of the separate habitats. The optimal processes are themselves two-type branching processes with random environments.

Let X1, X2, · ··, Xn be independent random variables such that ai ≦ Xi ≦ bi, i = 1,2,…n. A class of upper bounds on the probability P(S−ES ≧ nδ) is derived where S = Σf(Xi), δ > 0 and f is a continuous convex function. Conditions for the exponential convergence of the bounds are discussed.

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.

An anticipation process is a continuous-time Markov process, having an element of deterministic drift in its transition structure. The theory of such processes, developed here from an elementary (Chapman-Kolmogorov) point of view, draws together a number of threads of theoretical and applied probability.

Consider the following path, Zn(w), of a Galton-Watson process in reverse. The probabilities that ZN–n = j given ZN = i converge, as N → ∞ to a probability function of a Markov process, Xn, which I call the ‘reverse process’. If the initial state is 0, I require that the transition probabilities be the limits given not only ZN = 0 but also ZN–1 > 0. This corresponds to looking at a Galton-Watson process just prior to extinction. This paper gives the n-step transition probabilities for the reverse process, a stationary distribution if m ≠ 1, and a limit law for Xn/n if m = 1 and σ2 < ∞. Two related results about Zcn, 0 < c < 1, for Galton-Watson processes conclude the paper.

Let W (s, t), s, t ≧ 0, be the two-parameter Yeh–Wiener process defined on the first quadrant of the plane, that is, a Gaussian process with independent increments in both directions. In this paper, a lower bound for the distribution of the supremum of W (s, t) over a rectangular region [0, S]×[0, T], for S, T > 0, is given. An upper bound for the same was known earlier, while its exact distribution is still unknown.

It is well-known that the transition matrix of a reversible Markov process can have only real eigenvalues. An example is constructed which shows that the converse assertion does not hold. A generalised notion of reversibility is proposed, ‘dynamic reversibility’, which has many of the implications for the form of the transition matrix of the classical definition, but which does not exclude ‘circulation in state-space’ or, indeed, periodicity.

Let {X(t),t ≧0} be a process with stationary independent increments which is stochastically continuous with right-continuous paths and normalized so that X(0)=0. Let Z1(t) = X(t), Z2(t) = sup0≦s≦tX(s) and Z3 (t) = largest positive jump of X in (0, t] if there is one; = 0 otherwise. Then for i = 1,2,3 and x > 0: limt↓0t—1P[Zi(t) > x] = M+(x) at all points of continuity of M+, the Lévy measure of X.

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.

Let X(t), N(t) respectively denote the number of cells alive at t and the total number of cells born by t in a critical age-dependent Bellman-Harris branching process.

The asymptotic behavior of the conditional moments, for 0 < α ≦ 1, E(Nn(αt) | X(t) > 0), E(Nn(t) |X(αt) > 0), is obtained.

The problem considered is that of measuring the velocity of a signal, constituted by a plane wave, from measurements at a number of recorders receiving noise as well as signal. The asymptotic properties of the estimates are considered under rather general conditions on the noise and signal processes.

Time-reversibility is defined for a process X(t) as the property that {X(t1), …, X(tn)} and {X(– t1), …, X(– tn)} have the same joint probability distribution. It is shown that, for discrete mixed autoregressive moving-average processes, this is a unique property of Gaussian processes.

It is shown that methods used by the author to approximate limit probabilities for Markov processes from their Q-matrices extend to semi-Markov processes. The limit probabilities for semi-Markov processes can be approximated using only truncations of the embedded Markov chain transition matrix and the vector of mean holding times.

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.

This paper is an attempt to interpret and extend, in a more statistical setting, techniques developed by D. L. Snyder and others for estimation and filtering for doubly stochastic point processes. The approach is similar to the Kalman-Bucy approach in that the updating algorithms can be derived from a Bayesian argument, and lead ultimately to equations which are similar to those occurring in stochastic approximation theory. In this paper the estimates are derived from a general updating formula valid for any point process. It is shown that almost identical formulae arise from updating the maximum likelihood estimates, and on this basis it is suggested that in practical situations the sequence of estimates will be consistent and asymptotically efficient. Specific algorithms are derived for estimating the parameters in a doubly stochastic process in which the rate alternates between two levels.

For the diffusion branching process, we consider a method of inference that is essentially sequential in nature. The method allows us to simplify the natural sufficient statistics involved, and we are able to get their distributions quite easily by translating our problem into a standard problem in Brownian motion. Under certain circumstances, we are left with a complete sufficient statistic whose distribution belongs to an exponential family, and can therefore derive minimum variance unbiased estimators, etc.

Let Y, X1, X2, …, Xn be a set of n + 1 independently and uniformly distributed random variables on the interval (0, 1). The distribution of the length of the minimum path starting at Y which covers the other n points is derived. The solution is interesting in that it involves finding the distribution of an order statistic of a function of order statistics.

The joint distribution of the time since last exit, and the time until next entrance, into a unique boundary point is given in Formula (1) below. The boundary point may be replaced by a regenerative phenomenon.