On Stein's method for stochastically monotone single-birth chains

TL;DR

This paper develops Stein's method for analyzing the convergence and distributional approximation of stochastically monotone single-birth Markov chains, providing bounds on convergence rates and distributional distances.

Contribution

It introduces bounds on solutions to Poisson's equation for single-birth chains under stochastic monotonicity, enabling new convergence and comparison results.

Findings

Derived bounds on convergence to stationarity.

Provided methods to bound total variation distance between stationary distributions.

Applied results to specific classes of single-birth chains.

Abstract

We discuss Stein's method for approximation by the stationary distribution of a single-birth Markov chain, in conjunction with stochastic monotonicity and similar assumptions. We use bounds on the increments of the solution of Poisson's equation for such a process. Applications include rates of convergence to stationarity, and bounding the total variation distance between the stationary distributions of two Markov chains in the case where one transition matrix dominates the other.