Siegmund Duality for Markov Chains on Partially Ordered State Spaces

Abstract

For a Markov chain on a finite partially ordered state space, we show that its Siegmund dual exists if and only if the chain is Möbius monotone. This is an extension of Siegmund's result for totally ordered state spaces, in which case the existence of the dual is equivalent to the usual stochastic monotonicity. Exploiting the relation between the stationary distribution of an ergodic chain and the absorption probabilities of its Siegmund dual, we present three applications: calculating the absorption probabilities of a chain with two absorbing states knowing the stationary distribution of the other chain; calculating the stationary distribution of an ergodic chain knowing the absorption probabilities of the other chain; and providing a stable simulation scheme for the stationary distribution of a chain provided we can simulate its Siegmund dual. These are accompanied by concrete examples: the gambler's ruin problem with arbitrary winning/losing probabilities; a non-symmetric game; an extension of a birth and death chain; a chain corresponding to the Fisher–Wright model; a non-standard tandem network of two servers, and the Ising model on a circle. We also show that one can construct a strong stationary dual chain by taking the appropriate Doob transform of the Siegmund dual of the time-reversed chain.

Generalized Gambler’s Ruin Problem: Explicit Formulas via Siegmund Duality

“…Recently, [HM14] considered dualities for Markov chains on partitions and sets. In [Lor15] we show that Siegmund dual exists if and only if chain is Möbius monotone, the connections with Strong Stationary Duality (consult [DF90]) is also given therein. Let us mention at this point that for non-linear ordering Möbius and stochastic monotonicities are, in general, different.…”

Absorption time and absorption probabilities for a family of multidimensional gambler models

“…In particular, for a subclass of the multidimensional chains, constructed from one-dimensional birth and death chains, the winning probabilities are given in (3). The main tool for showing winning probabilities is the Siegmund duality defined for partially ordered state spaces, exploiting the results from [12].…”

“…The existence of a Siegmund dual for linearly ordered state space requires stochastic monotonicity of a chain. Recently, in [12] we provided if and only if conditions for existence of Siegmund dual for partially ordered state spaces (roughly speaking, the Möbius monotonicity is required). In this paper, we exploit this duality defined for a coordinate-wise partial ordering.…”

Absorption time and absorption probabilities for a family of multidimensional gambler models

“…In particular, for a subclass of multidimensional chains, constructed from one-dimensional birth and death chains, the winning probabilities are given in (1.3). The main tool for showing winning probabilities is the Siegmund duality defined for partially ordered state spaces, exploiting the results from Lorek (2018).…”

“…The existence of a Siegmund dual for a linearly ordered state space requires stochastic monotonicity of the chain. Recently, in Lorek (2018) we provided if and only if conditions for the existence of the Siegmund dual for partially ordered state spaces (roughly speaking, the Möbius monotonicity is required). In this paper, we exploit this duality defined for a coordinate-wise partial ordering.…”