The equilibrium probability distribution over the set of absorbing states of a reducible Markov chain is specified a priori and it is required to obtain the constrained sub-space or feasible region for all possible initial probability distributions over the set of transient states. This is called the inverse problem. It is shown that a feasible region exists for the choice of equilibrium distribution. Two different cases are studied: Case I, where the number of transient states exceeds that of the absorbing states and Case II, the converse. The approach is via the use of generalised inverses and numerical examples are given.

Limit laws and limiting diffusions are obtained for critical branching processes, either Galton-Watson or age-dependent, conditioned on extinction in the interval (T, cT], 1 < c, as T→∞, and also as T→∞ and c ↓ 1.

Particles enter a finite-state system and move according to independent sample paths from a semi-Markov process. Strong limit theorems are developed for the ratio of the flow of particles from states i to j and the flow out of When the cumulative arrival of particles into the system up to time t, A (t) ∼ λtα, then a.s. When A (t)∼ λekt, then the flow between states must be normalized by the Laplace–Stieltjes transform of the conditional holding time distribution, in order to make the ratio an unbiased estimator of ρij.

Kaufmann's exact characterization of the partition function for the classical Ising model is used to obtain limit theorems for the sample correlation between nearest neighbours in the non-critical case. This provides a basis for the asymptotic testing and estimation (by confidence intervals) of the correlation between nearest neighbours.

The joint probability density function of the occupation time of a three-state stochastic process with constant transition matrix: aij ≠ 0, i ≠ j, i, j = 1, 2, 3 but αii = 0, i = 1, 2, 3 is studied.

A generalization of a theorem of Geman and Horowitz concerning random measures is obtained. Their theorem in turn is used to extend a well-known result of Lumley, which asserts that for homogeneous turbulence in an incompressible fluid the probability distributions of the Eulerian and Lagrangian velocities coincide with each other at all times.

The classical Crofton formula is generalized to the case of thick sections, and the corresponding corrective formulae are given.

This paper provides a rigorous foundation for the second-order analysis of stationary point processes on general spaces. It illuminates the results of Bartlett on spatial point processes, and covers the point processes of stochastic geometry, including the line and hyperplane processes of Davidson and Krickeberg. The main tool is the decomposition of moment measures pioneered by Krickeberg and Vere-Jones. Finally some practical aspects of the analysis of point processes are discussed.

A martingale approach to a dynamic program with general state and action spaces is taken. Several necessary and sufficient conditions are given for a policy to be optimal. The results comprehend and modify different criteria of optimality given for dynamic programming problems. Finally, two applications are stated.

Let {W(t), 0 ≦ t < ∞} be the standard Wiener process. The main purpose of this paper is to present ways of obtaining probabilities of Wiener paths crossing certain curves on various intervals. The results are extended to other kinds of Gaussian processes.

Formulae relating properties of a two-phase structure to properties of a section through that structure are well known in stereology. An exact description of the error introduced into these formulae when the projection of a thick slice is observed rather than a true section (i.e. a slice of zero thickness) is presented for a model of opaque particles distributed according to a homogeneous Poisson point process in En.

A waiting time without memory, or age-independent residual life-time, is a positive-valued random variable T with the property that for any x, y > 0, given that T >x, the conditional probability of T >x +y is the same as the unconditional probability of T >y; in other words, the physical process operates as if it has no memory concerning the successive occurrences of a certain event. The paper investigates the consequences of defining the property of lack of memory on more general time-domains than the positive reals. As a side issue, there is discussion of a stochastic variation of Cauchy's functional equation.

A class of limit theorems involving asymptotic normality is derived for stationary processes whose spectral density has a singular behavior near frequency zero. Generally these processes have ‘long-range dependence’ but are generated from strongly mixing processes by a fractional integral or derivative transformation. Some related remarks are made about random solutions of the Burgers equation.

For an n-dimensional random field X(t) we define the excursion set A of X(t) by A = [t ∊ S: X(t) ≧ u] for real u and compact S ⊂ Rn. We obtain a generalisation of the number of upcrossings of a one-dimensional stochastic process to random fields via a characteristic of the set A related to the Euler characteristic of differential topology. When X(t) is a homogeneous Gaussian field satisfying certain regularity conditions we obtain an explicit formula for the mean value of this characteristic.

A basic result in the theory of total positivity is that the convolution of any two Pólya frequency sequences is again a Pólya frequency sequence. The like result for binomial convolution, associated with exponential generating functions, is proved. This and similar results are used to obtain an upper bound on the probability that the union of independent random subsets of a finite set N is all of N. Parallels from the theory of reliability involving sums of random variables with increasing failure rates are noted.

Necessary and sufficient conditions are presented under which the point processes equivalent to order statistics of n i.i.d. random variables or superpositions of n i.i.d. renewal processes converge to a non-degenerate limiting process as n approaches infinity. The limiting process must be one of three types of non-homogeneous Poisson process, one of which is the Weibull process. These point processes occur as failure-time models in the reliability theory of repairable systems.

For stationary isotropic random functions on a Euclidean space, we characterize and compare the mean values of certain geometric measures of the smoothness of realizations. In particular we examine mean properties of the contours and gradients of the random function, and the effect of local averaging on smoothness in special cases.

Two types of limit theorems are proved for processes of randomly displaced regular events. Firstly, as the displacements tend to infinity, the counting process is shown to converge weakly to a Poisson process and secondly, as the interval between events tends to zero, convergence of the finite-dimensional distributions of the associated storage process to a diffusion is proved.

Let Gn (x) be the distribution of the nth successive maximum of a random walk on the real line. Under conditions typical for complete exponential convergence, the decay of Gn (x) – limn→∞ Gn(x) is asymptotically equal to H(x) γn n–3/2 as n → ∞where γ < 1 and H(x) a function solely depending on x. For the case of drift to + ∞, G∞(x) = 0 and the result is new; for drift to – ∞we give a new proof, simplifying and correcting an earlier version in [9].

We extend some results of Hammersley and Welsh concerning first-passage percolation on the two-dimensional integer lattice. Our results include: (i) weak renewal theorems for the unrestricted reach processes; (ii) an L1-ergodic theorem for the unrestricted first-passage time from (0, 0) to the line X = n; and (iii) weakening of the boundedness restrictions on the underlying distribution in Hammersley and Welsh's weak renewal theorems for the cylinder reach processes.