Monotonicity equivalence and synchronizability for a system of probability distributions

Abstract

A system $(P_α: α\in\mathcal{A})$ 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 $X_α$'s marginally distributed as $P_α$. It is called realizably monotone if $X_α\le X_β$ in $\mathcal{S}$ whenever $α\leβ$ in $\mathcal{A}$. Such a system necessarily is stochastically monotone, that is, it satisfies $P_α\preceq P_β$ in stochastic ordering whenever $α\le β$. 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 $(X_α: α\in\mathcal{A})$ explicitly. We also show that synchronizability is necessary for monotonicity equivalence when $\mathcal{S}$ is in Class W.