A system $(P_\alpha: \alpha\in\mathcal{ A})$ of probability distributions on a partially ordered set (poset) $\mathcal{ S}$ indexed by another poset $\mathcal{ A}$ can be realized by a system of $\mathcal{ S}$-valued random variables $X_\alpha$'s marginally distributed as $P_\alpha$. It is called realizably monotone if $X_\alpha\le X_\beta$ in $\mathcal{ S}$ whenever $\alpha\le\beta$ in $\mathcal{ A}$. Such a system necessarily is stochastically monotone, that is, it satisfies $P_\alpha\preceq P_\beta$ in stochastic ordering whenever $\alpha \le \beta$. It has been known exactly when these notions of monotonicity are equivalent except for a certain subclass of acyclic posets, called Class W. In this paper we introduce inverse probability transforms and synchronizing bijections recursively when $\mathcal{ S}$ is a poset of Class W and $\mathcal{ A}$ is synchronizable, and validate monotonicity equivalence by constructing $(X_\alpha: \alpha\in\mathcal{ A})$ explicitly. We also show that synchronizability is necessary for monotonicity equivalence when $\mathcal{ S}$ is in Class W.
Categories: math.PR