A Correlation Inequality for Markov Processes in Partially Ordered State Spaces

Let E be a finite partially ordered set and Mₚ the set of probability measures in E giving a positive correlation to each pair of increasing functions on E. Given a Markov process with state space E whose transition operator (on functions) maps increasing functions into increasing functions, let Uₜ be the transition operator on measures. In order that UₜMₚ ⊂ Mₚ for each t 0, it is necessary and sufficient that every jump of the sample paths is up or down.