The Prelimit Generator Comparison Approach of Stein’s Method

Anton Braverman

doi:10.1287/stsy.2021.0085

Abstract

This paper uses the generator comparison approach of Stein’s method to analyze the gap between steady-state distributions of Markov chains and diffusion processes. The “standard” generator comparison approach starts with the Poisson equation for the diffusion, and the main technical difficulty is to obtain bounds on the derivatives of the solution to the Poisson equation, also known as Stein factor bounds. In this paper we propose starting with the Poisson equation of the Markov chain; we term this the prelimit approach. Although one still needs Stein factor bounds, they now correspond to finite differences of the Markov chain Poisson equation solution rather than the derivatives of the solution to the diffusion Poisson equation. In certain cases, the former are easier to obtain. We use the [Formula: see text] model as a simple working example to illustrate our approach.

Highlights

Recent years have seen a growing use of the generator comparison approach of Stein’s method to establish rates of convergence for steady-state diffusion approximations of Markov chains
The “standard” generator comparison approach starts with the Poisson equation for the diffusion, and the main technical difficulty is to obtain bounds on the derivatives of the solution to the Poisson equation, known as Stein factor bounds
In this paper we propose starting with the Poisson equation of the Markov chain; we term this the prelimit approach

Summary

Introduction

Recent years have seen a growing use of the generator comparison approach of Stein’s method to establish rates of convergence for steady-state diffusion approximations of Markov chains. This excludes diffusions with reflecting boundary conditions, such as reflecting Brownian motions that appear as heavy-traffic limits for networks of single-server queueing systems Another approach to getting derivative bounds was proposed in Gurvich (2014a), where the author used a priori Schauder estimates from PDE theory to bound the derivatives of fh∗ (x) in terms of fh∗ (x) and h(x). In the case of a diffusion with a reflecting boundary, the complexity of the PDE machinery used makes it nontrivial to trace how the a priori Schauder estimates depend on the primitives of the diffusion process Most recently, another approach to getting derivative bounds based on Bismut’s formula from Malliavin calculus was proposed in Fang et al (2019). The authors required the diffusion coefficient to be constant, and the assumptions imposed on the drift were similar to those in Mackey and Gorham (2016)

Notation

The Prelimit Generator Comparison Approach

Convergence-Determining Classes

Misalignment of Diffusion Synchronous Couplings

Conclusion