Strong stationary duality for Möbius monotone Markov chains

Abstract

For Markov chains with a finite, partially ordered state space, we show strong stationary duality under the condition of Möbius monotonicity of the chain. We give examples of dual chains in this context which have no downwards transitions. We illustrate general theory by an analysis of nonsymmetric random walks on the cube with an interpretation for unreliable networks of queues.

Antiduality and Möbius monotonicity: generalized coupon collector problem

“…which is a stochastic matrix if and only if each entry of C −1 ← PC is non-negative, in other words we say that ← P is Möbius monotone. This way we proved the main part of Theorem 2 of [30]. We include it here, since this is a little bit different (matrix-form) proof.…”

“…In [8] authors give a recipe for a dual on the same state space E * = E provided that a time reversed chain ← X is stochastically monotone with respect to total ordering. In [30] we give an extension of this result to state spaces which are only partially ordered by ⪯. Then, provided that the time reversed chain ← X is Möbius monotone (plus some conditions on the initial distribution), we give a formula for a sharp SSD on the same state space E * = E.…”

“…TITLE WILL BE SET BY THE PUBLISHER Theorem 3.5 (Theorem 2 of [30]). Let X ∼ (ν, P) be an ergodic Markov chain on a finite state space E = {e 1 , .…”

“…TITLE WILL BE SET BY THE PUBLISHER Remark 4.9. In [30] we considered the chain on E = {0, 1} d with transition matrix P 3 given by…”

On strong stationary times and approximation of Markov chain hitting times by geometric sums

“…Recall that a strong stationary time, T , for X is a randomized stopping time such that X T ∼ π, and X T is independent of T . We refer the reader to [1] and [4] for background on strong stationary times, and to [5], [8] and references therein for more recent developments.…”