Annals of Probability

Stochastic Monotonicity and Realizable Monotonicity

James Allen Fill and Motoya Machida

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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

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

First available in Project Euclid: 21 December 2001

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E05: Distributions: general theory
Secondary: 06A06: Partial order, general 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 05C38: Paths and cycles [See also 90B10]

Realizable monotonicity stochastic monotonicity monotonicity equivalence perfect sampling partially ordered set Strassen's theorem marginal problem inverse probability transform cycle rooted tree


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

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