Siegmund duality for Markov chains on partially ordered state spaces

Pawel Lorek

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 equiva-
lent 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.