Source: Ann. Probab. Volume 29, Number 2
(2001), 938-978.
We explore and relate two notions of monotonicity, stochastic and
realizable, for a system of probability measures on a common finite partially
ordered set (poset) $\mathcal{S}$ when the measures are indexed by another
poset $\mathcal{A}$. We give counterexamples to show that the two notions are
not always equivalent, but for various large classes of $\mathcal{S}$ we also
present conditions on the poset $\mathcal{A}$ that are necessary and sufficient
for equivalence. When $\mathcal{A} = \mathcal{S}$ , the condition that the
cover graph of $\mathcal{S}$ have no cycles is necessary and sufficient for
equivalence. This case arises in comparing applicability of the perfect
sampling algorithms of Propp and Wilson and the first author of the present
paper.
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