RealizableMonotonicity and Inverse Probability Transform

A system (Pα : α ∈ A) of probability measures on a common state space S indexed by another index set A can be “realized” by a system (Xα : α ∈ A) of S-valued random variables on some probability space in such a way that each Xα is distributed as Pα. Assuming that A and S are both partially ordered, we may ask when the system (Pα : α ∈ A) can be realized by a system (Xα : α ∈ A) with the monotonicity property that Xα ≤ Xβ almost surely whenever α ≤ β. When such a realization is possible, we call the system (Pα : α ∈ A) “realizably monotone.” Such a system necessarily is stochastically monotone, that is, satisfies Pα ≤ Pβ in stochastic ordering whenever α ≤ β. In general, stochastic monotonicity is not sufficient for realizable monotonicity. However, for some particular choices of partial orderings in a finite state setting, these two notions of monotonicity are equivalent. We develop an inverse probability transform for a certain broad class of posets S, and use it to explicitly construct a system (Xα : α ∈ A) realizing the monotonicity of a stochastically monotone system when the two notions of monotonicity are equivalent.