This paper treats stopping problems on Markov chains in which the OLA (one-step look ahead) policy is optimal. Its associated optimal value can be explicitly expressed by a potential for a charge function of the difference between the immediate reward and the one-step-after reward. As an application to the best choice problem, we shall obtain the value of three problems: the classical secretary problem, a problem with a refusal probability and a problem with a random number of objects.

One-dimensional, periodic and annihilating systems of Brownian motions and random walks are defined and interpreted in terms of sizeless particles which vanish on contact. The generating function and moments of the number pairs of particles which have vanished, given an arbitrary initial arrangement, are derived in terms of known two-particle survival probabilities. Three important special cases are considered: Brownian motion with the particles initially (i) uniformly distributed and (ii) equally spaced on a circle and (iii) random walk on a lattice with initially each site occupied. Results are also given for the infinite annihilating particle systems obtained in the limit as the number of particles and the size of the circle or lattice increase. Application of the results to the theory of diffusion-limited reactions is discussed.

Stereological analysis of d-dimensional particles of ellipsoidal shape based on lower-dimensional sections through the particles is discussed. It is proved that the non-void intersections between three parallel hyperplanes and an ellipsoid uniquely determine the ellipsoid, and based on this fact we may reconstruct ellipsoids from sectional information. Combining this reconstruction with a new sampling procedure we obtain a useful tool for non-parametric stereological analysis of particle aggregates of ellipsoids. Finally, parametric models for ellipsoids which are mathematically convenient for the present set up are introduced and discussed.

Let {X(t): t ≧ 0} be a regenerative process with excursion process {Y(u): 0 ≦ u < T, T} and state space . Let An be a sequence of sets in such that pn=P(Y(u)∊An for some 0≦ u < T) → 0. Let Vn be the hitting time of An for the process X. This paper gives a variety of conditions on the excursion process Y to obtain limit theorems for Vn. Apart from obtaining an exponential limit in the positive recurrent case, i.e. ET < x, some non-exponential limits are obtained in the null recurrent case. The results are illustrated via the age process.

Some simple methods of estimating inbreeding from tests on a random sample of individuals are discussed. These are mostly iterative procedures, converging to the maximum likelihood estimate, with algorithms for finding standard errors. But there are problems of interpretation, both from the statistical and biological points of view.

Problems associated with setting up a probability model which generates the quantum theoretical probabilities for the two spin 1/2 particle system are examined. Arguments which claim to show that such a model cannot be constructed within classical probability theory under the assumption of local singlet states are also considered. It is shown that the model then in question is not a probability model in the sense that term is used elsewhere in science. An alternative model is proposed and its bearing on the Einstein-Bohr debate is briefly discussed.

Exact expressions are developed for the nth order autocovariance structure of the telegraph wave, the integral of which defines a biased correlated random walk.

This paper presents new methods for selecting the best population value under the classical secretary problem sequential selection rules. Unlike the classical case, the possible population arrival sequences considered here are not necessarily equally likely. When arrival orders in the sequence are rank dependent, the performance of the optimal solution to the standard problem (totally random arrival order assumed) is very poor, compared with procedures designed to guard against non-uniform arrival orders.

A probability density function important in the Poisson Dirichlet process of population genetics is studied. An accurate computational algorithm is given for this density and for the marginal distributions of the points in the Poisson Dirichlet process. The distribution of the maximal point of the process is tabulated. Rational polynomial approximations in θ, the mutation parameter, are found for the expected values of the first three maximal points.

A sufficient condition is derived for the existence of a strictly stationary solution of the general bilinear time series equations. The condition is shown to reduce to the conditions of Pham and Tran (1981) and Bhaskara Rao et al. (1983) in the special cases which they consider. Under the condition specified, a solution is constructed which is shown to be causal, stationary and ergodic. It is moreover the unique causal solution and the unique stationary solution of the defining equations. In the special case when the defining equations contain no non-linear terms, our condition reduces to the well-known necessary and sufficient condition for existence of a causal stationary solution.

We consider a system where jobs are processed by parallel machines. The processing times are exponentially distributed. An essential feature is that the assignment of the jobs to the machines is decided before the system starts to work. We consider both the flow time and the makespan. In the case of the flow time we allow both the machines and the jobs to be non-homogeneous. The optimization is by minimizing the flow time in the sense of stochastic order and the optimal assignment is obtained for this case. The case of the makespan is harder. We consider the expected makespan and as a partial solution we prove an optimality result for the case where there are two non-homogeneous machines and the jobs are homogeneous. It turns out that the optimal assignment can be expressed by using a quantile of a binomial distribution.

A supplemented GSMP (generalized semi-Markov process) is a useful stochastic process for discussing fairly general queues including queueing networks. Although much work has been done on its insensitivity property, there are only a few papers on its general properties. This paper considers a supplemented GSMP in a general setting. Our main concern is with a system of Laplace–Stieltjes transforms of the steady state equations called the basic equations. The basic equations are derived directly under the stationary condition. It is shown that these basic equations with some other conditions characterize the stationary distribution. We mention how to get a solution to the basic equations when the solution is partially known or inferred. Their applications to queues are discussed.

The notion of a quantum random walk in discrete time is formulated and the passage to a continuous time diffusion limit is established. The limiting diffusion is described in terms of solutions of certain quantum stochastic differential equations.

A derivation of a law of large numbers for the highest-scoring matching subsequence is given. Let Xk, Yk be i.i.d. q=(q(i))i∊S letters from a finite alphabet S and v=(v(i))i∊S be a sequence of non-negative real numbers assigned to the letters of S. Using a scoring system similar to that of the game Scrabble, the score of a word w=i1 · ·· im is defined to be V(w)=v(i1) + · ·· + v(im). Let Vn denote the value of the highest-scoring matching contiguous subsequence between X1X2 · ·· Xn and Y1Y2· ·· Yn. In this paper, we show that Vn/K log(n) → 1 a.s. where K ≡ K(q,v). The method employed here involves ‘stuttering’ the letters to construct a Markov chain and applying previous results for the length of the longest matching subsequence. An explicit form for β ∊Pr(S), where β (i) denotes the proportion of letter i found in the highest-scoring word, is given. A similar treatment for Markov chains is also included.

Implicit in these results is a large-deviation result for the additive functional, H ≡ Σn < τv(Xn), for a Markov chain stopped at the hitting time τ of some state. We give this large deviation result explicitly, for Markov chains in discrete time and in continuous time.

We present an identity which can sometimes express the joint distribution of several linear combinations of uniform spacings as a sum of simpler distributions. Examples are given to show that this identity is useful in the exact computation of probabilities and expectations which arise in testing for uniformity. The identity is also used to provide a new derivation of the distribution of a linear combination of spacings.

Some concepts of dependence have recently been introduced by Ebrahimi (1987) to explore the structural properties of the hitting times of bivariate processes. In this framework, the special case of univariate processes has curious features. New properties are derived for this case. Some applications to sequential inference and inequalities for Brownian motion and new better than used (NBU) processes are also provided.

Two service schemes for an M/M/1 time-sharing system with a limited number of service positions are studied. Both schemes possess the equilibrium properties of symmetric queues; however, in the first one, a preempted job is placed at the end of the waiting line; while in the second one, it is placed at the head of the line. Methods for calculating the Laplace transforms and moments of the response times are presented. The variances of the response times are then compared numerically to indicate that the first scheme is superior to the second scheme. It is also indicated that in both cases the response time variance decreases when the number of service positions increases.

A multivariate distribution for describing the life-lengths of the components of a system which operates in an environment that is different from the test bench environment has been proposed by Lindley and Singpurwalla (1986). In this paper, the properties of the reliability function of such a system are studied and comparisons made with the reliability function obtained under the assumption of independence. It is interesting to note that the reliability function of parallel redundant systems whose components share a common unknown environment cannot be characterized by any of the well-known classes of distributions that have been proposed in the mathematical theory of reliability. This observation suggests the need for defining a new class of failure distributions. A formula for making Bayesian inferences for the reliability function is also given.

The arc-sine laws form one of the cornerstones of classical one-dimensional fluctuation theory. In higher dimensions, knowledge of fluctuation theory remains a great deal less complete. Motivated by this, we consider higher-dimensional analogues of the classical arc-sine laws.

Five distributions on the non-negative integers are given which use the orthogonal polynomials of Hermite, Legendre, Laguerre and Chebyshev.