Quantitative Convergence Rates for Stochastically Monotone Markov Chains

TL;DR

This paper provides quantitative bounds on the convergence rates of stochastically monotone Markov chains, extending classical stability results and unifying them with total variation bounds.

Contribution

It introduces explicit convergence rate bounds for stochastically monotone Markov chains, enhancing understanding of their stability and ergodicity properties.

Findings

Quantitative bounds on distribution deviations are established.

Total variation bounds are recovered as a special case.

Results apply to a broad class of stochastically monotone Markov chains.

Abstract

For Markov chains and Markov processes exhibiting a form of stochastic monotonicity (larger states shift up transition probabilities in terms of stochastic dominance), stability and ergodicity results can be obtained using order-theoretic mixing conditions. We complement these results by providing quantitative bounds on deviations between distributions. We also show that well-known total variation bounds can be recovered as a special case.