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A stochastic process {X(t): 0 S t < =} with discrete state space
S c ~ is said to be stochastically increasing (decreasing) on an
interval T if the probabilities Pr{X(t) > i}, i E S, are
increasing (decreasing) with t on T. Stochastic 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, stochastic monotonicity becomes a more
tractable subject for analysis if the processes under consideration
are such that stochastic mono tonicity on an inter val 0 < t
< E implies stochastic 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 "stochastic 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 stochas
tically monotone as defined above, now depends on the initial state
distribution. Conditions on this distribution for stochastic mono
tonicity on the entire time axis to prevail were given too by
KEILSON and KESTER (1977).
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