A Unified Stability Theory for Classical and Monotone Markov Chains

TL;DR

This paper develops a unified stability framework for classical and monotone Markov chains by introducing a new metric based on partial stochastic dominance, bridging traditional and order-theoretic stability results.

Contribution

It introduces a complete metric over probability measures that generalizes total variation, enabling unified stability analysis for both classical and monotone Markov chains.

Findings

Generalizes stability results to partially ordered settings

Introduces a new metric based on partial stochastic dominance

Bridges classical and order-theoretic Markov chain stability theories

Abstract

This paper integrates two strands of the literature on stability of general state Markov chains: conventional, total variation based results and more recent order-theoretic results. First we introduce a complete metric over Borel probability measures based on partial stochastic dominance. We then show that many conventional results framed in the setting of total variation distance have natural generalizations to the partially ordered setting when this metric is adopted.