A study is made of the maximum, minimum and range on [0, t] of the integral process where S is a finite state-space Markov chain. Approximate results are derived by establishing weak convergence of a sequence of such processes to a Wiener process. For a particular family of two-state stationary Markov chains we show that the corresponding centered integral processes exhibit the Hurst phenomenon to a remarkable degree in their pre-asymptotic behaviour.

The method of invariant imbedding, used extensively in astrophysics and in neutron transport theory, is applied to the analysis of dams whose input rate is a finite state-space continuous-parameter Markov chain. Passage-time distributions are considered as functions of the dam capacity. When the release rate is constant the method yields first-order non-linear differential equations with initial conditions obtained from the easily derived behaviour of a dam with zero capacity. Solutions of these equations are given. Some possible correlation functions for the increments of the cumulative input process are examined when the underlying Markov chain is stationary. Finally the analysis is extended to deal with dams having content-dependent release-rate of the form