## Annals of Probability

### Stochastic Monotonicity and Realizable Monotonicity

#### Abstract

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) $\mathscr{S}$ when the measures are indexed by another poset $\mathscr{A}$. We give counterexamples to show that the two notions are not always equivalent, but for various large classes of $\mathscr{S}$ we also present conditions on the poset $\mathscr{A}$ that are necessary and sufficient for equivalence. When $\mathscr{A} = \mathscr{S}$ , the condition that the cover graph of $\mathscr{S}$ 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.

#### Article information

Source
Ann. Probab., Volume 29, Number 2 (2001), 938-978.

Dates
First available in Project Euclid: 21 December 2001

https://projecteuclid.org/euclid.aop/1008956698

Digital Object Identifier
doi:10.1214/aop/1008956698

Mathematical Reviews number (MathSciNet)
MR1849183

Zentralblatt MATH identifier
1015.60010

#### Citation

Fill, James Allen; Machida, Motoya. Stochastic Monotonicity and Realizable Monotonicity. Ann. Probab. 29 (2001), no. 2, 938--978. doi:10.1214/aop/1008956698. https://projecteuclid.org/euclid.aop/1008956698