A shastic process {X(t): 0 S t < =} with discrete state space S c ~ is said to be shastically increasing (decreasing) on an interval T if the probabilities Pr{X(t) > i}, i E S, are increasing (decreasing) with t on T. Shastic monotonicity is a basic structural property for process behaviour. It gives rise to meaningful bounds for various quantities such as the moments of the process, and provides the mathematical groundwork for approximation algorithms. Obviously, shastic monotonicity becomes a more tractable subject for analysis if the processes under consideration are such that shastic mono tonicity on an inter­ val 0 < t < E implies shastic monotonicity on the entire time axis. DALEY (1968) was the first to discuss a similar property in the context of discrete time Markov chains. Unfortunately, he called this property "shastic monotonicity", it is more appropriate, however, to speak of processes with monotone transition operators. KEILSON and KESTER (1977) have demonstrated the prevalence of this phenomenon in discrete and continuous time Markov processes. They (and others) have also given a necessary and sufficient condition for a (temporally homogeneous) Markov process to have monotone transition operators. Whether or not such processes will be shas­ tically monotone as defined above, now depends on the initial state distribution. Conditions on this distribution for shastic mono tonicity on the entire time axis to prevail were given too by KEILSON and KESTER (1977).