This paper describes the operator mapping a renewal-interval distribution into its associated stationary-excess distribution. This operator is monotone for some kinds of stochastic order, but not for the usual stochastic order determined by the expected value of all non-decreasing functions. Conditions for a renewal-interval distribution to be larger or smaller than its associated stationary-excess distribution for several kinds of stochastic order are determined in terms of familiar notions of ageing. Convergence results are also obtained for successive iterates of the operator, which supplement Harkness and Shantaram (1969), (1972) and van Beek and Braat (1973).

The paper gives the mean-time and variance-time functions of responses arising from a general type of interaction between two stationary point processes which are independent of each other. The mean-time function is obtained exactly and explicitly, thereby much improving on earlier work. A Laplace-transformed version of the variance-time function is derived under the general assumptions, and new results for the asymptotic slope and intercept are obtained when one of the point processes is renewal. Attention is drawn to the difficulties of dealing with non-renewal and non-regenerative point processes.

A recursion formula is obtained by rearranging Lawrance's (1973) result concerning the superposition of independent stationary point processes for which there exist joint probability density functions for the intervals between successive points. When these component point processes are identically distributed, the formula can in principle be inverted to describe their probability structure given that of the superposition process.

Additional theory important in the design and evaluation of a periodic screening program is presented. The distributions of doubly bounded preclinical state forward and backward recurrence times of the stochastic screening model are derived. Properties of these distributions are used to obtain an expression for estimating the mean lead time gained by screening, an alternative theoretical expression for the mean lead time and an equation for calculating an estimate of the probability of detecting preclinical disease.

A stochastic model for a periodic screening program is presented in which the natural history of a chronic disease is assumed to follow a progressive path from a preclinical state to a clinical state. The sampling of preclinical state sojourn times by screening examinations generates bounds on the preclinical state recurrence times. The distribution of the bounded forward recurrence time is derived and used to obtain the distribution and mean of the lead time, and relationships for calculating the proportion of preclinical cases detected. These expressions are derived in terms of the preclinical state sojourn-time distribution and adjustible parameters important in the design of a periodic screening program.

A bivariate point process consisting of an original stationary point process and its random translation is considered. Westcott's method is applied to show that if the original point process is ergodic then the bivariate point process is also ergodic. This result is applied to an identification problem of the displacement distribution. It is shown that if the spectrum of the original process is the real line then the displacement distribution is identifiable from almost every sample realisation of the bivariate process.