Abstract
Let $E$ be a finite partially ordered set and $M_p$ 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 $U_t$ be the transition operator on measures. In order that $U_tM_p \subset M_p$ for each $t \geqq 0$, it is necessary and sufficient that every jump of the sample paths is up or down.
Citation
T. E. Harris. "A Correlation Inequality for Markov Processes in Partially Ordered State Spaces." Ann. Probab. 5 (3) 451 - 454, June, 1977. https://doi.org/10.1214/aop/1176995804
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