Simple definitions and derivations of elementary properties are given for the various intensities and Palm-Khinchin functions associated with a multivariate point process.

An asymptotic expression for an absorption probability for the Ornstein-Uhlenbeck process is presented along with an application of the result to a problem in optional stopping. The relation of this result to the asymptotic behavior of a weighted Kolmogorov-Smirnov statistic is also discussed. Sweet and Hardin (1970) derive an exact solution (not in closed form) for this same problem.

Strassen-type law of the iterated logarithm for Brownian sheets presented by Pyke [7] is proved by using recent results of Kuelbs and Lepage [4]: the law of the iterated logarithm for Brownian motion in a Banach space and some applications are given.

In this paper models for diffusion in one dimension are obtained which are based on correlated random walks. The equations for diffusion with drift can be transformed into the equations for diffusion without drift (and conversely) by the transformations of Special Relativity Theory. The relationship of these equations to Maxwell's equations for electromagnetic phenomena is discussed.

Examples are given of point processes that are non-stationary but have stationary forward recurrence time distributions. They are obtained by modification of stationary Poisson and renewal processes.

We show that there exists a pair of Poisson processes which are not stochastically independent, and whose sum is still a Poisson process.

Kerridge introduced a measure known as inaccuracy for complete probability distributions which is the generalisation of Shannon's entropy. In this paper we study a grouping property of the inaccuracy. Also we have established a coding theorem for personal codes by considering inaccuracy of order a and generalised mean length of order t under the condition .

This paper describes an accurate method of approximating the mean of the first-passage time distribution for an Ornstein-Uhlenbeck process with a single absorbing barrier. The accuracy of the approximation is demonstrated through some numerical comparisons.

The restoration of crystal lattices which have been jittered presents some problems that do not seem to have been previously studied. This is true especially in the multi-dimensional case. It is shown that a statistic suggested by Bartlett can be used for this purpose, and this is done both for Gaussian and general deformation mechanisms. The results are accompanied by some numerical experiments.

Herein is exposed a simplified analytic proof of formulas for the characteristic functions of ordered partial sums of mutually independent identically distributed random variables. This problem which we had raised and solved in 1952 by another method, has since been treated by several authors (see Wendel [6]), and recently by de Smit [4], who made use of a kind of Wiener-Hopf decomposition†. On the contrary our present as well as our previous proof essentially uses the explicit solution of a certain singular integral equation in a complex domain.

A discrete-time Markov process on the interval [0, 1] is considered. Sufficient conditions for the existence of a unique stationary limiting distribution are given.

The strong law of large numbers for independent and identically distributed random variables Xi, i = 1,2,3, …, with finite mean µ can be stated as, for any ∊ > 0, the number of integers n such that |n−1 Σi=1nXi − μ| > ∊, N(∊), is finite a.s. It is known, furthermore, that EN(∊) < ∞ if and only if EX12 < ∞. Here it is shown that if EX12 < ∞ then ∊2EN(∊)→ var X1 as ∊ → 0.

Simple bounds on service level and turnover velocity are obtained for a periodic-review inventory system with a stationary order-up-to-level stocking policy and no backordering. Exact computational formulas are given for Poisson demand. An illustrative numerical example is presented, and the application of these bounds to retail inventory analysis is discussed.

We give an elementary proof of the martingale characterization theorem for Poisson processes over the positive real half line. This theorem is due to Watanabe [8] in the case where the mean measure associated to the Poisson process is the Lebesgue measure.

Representing a rotation in three dimensions by a unit tensor quaternion and supposing that the errors in the measurements of a number of vector directions follow Fisher's distribution, the maximum likelihood estimator of a rotation is obtained. It is shown that the natural Haar measure for three-dimensional rotations is mapped 1:2 onto the natural Haar measure for random directions in 4-space. The natural Haar measure for rotations in 4-space is also mappable onto the product measure of the Haar measures for two separate random directions in 4-space.

The central limit theorem for ergodic stationary processes obtained by Gordin is shown to hold for general stationary processes. In this case, the limit law is a mixture of normals.

For dependent probability systems of m events partially specified by the quantities Sv, the sum of the probabilities of each of the combinations of v events, v = 1, 2, …, 2 ≦ k < m; this paper develops the most stringent upper and lower bounds on P1, the probability of the union of the m events; and on P[m], the probability of the simultaneous occurrence of the m events.

Recent characterizations of probability measures by use of conditional expectations have included many probability measures defined on subsets of the real line. This paper contains results which allow similar characterizations by use of conditional expectations but are valid for probability measures defined on subsets of an arbitrary space. In the special case of a measure defined on subsets of R' the result is not always as sharp as known results, but these theorems can be applied to measures defined on subsets of Rn with n ≧ 1.

The water level in a reservoir is modelled as a controlled diffusion process on a compact interval of the real line. The problem is to control the water discharge rate so as to minimise the expected costs, which depend upon the histories of the water levels and release rates. The form of the optimal control is studied for two general classes of reservoir control problems.

The trajectory of a car traveling at a constant speed on an idealized infinite highway can be viewed as a straight line in the time-space plane. Entry times are governed by a Poisson process with intensity parameter A leading to all trajectories as random lines in a plane. The Poisson distribution of number of encounters of cars on the highway is developed through random line models and non-homogeneous Poisson fields, and its parameter, which depends on the specific random measure employed, is obtained explicitly.