The Prelimit Generator Comparison Approach of Stein’s Method

Anton Braverman*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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 M=M=1 model as a simple working example to illustrate our approach.

Original languageEnglish (US)
Pages (from-to)181-204
Number of pages24
JournalStochastic Systems
Volume12
Issue number2
DOIs
StatePublished - Jun 2022

Keywords

  • Markov chain
  • Stein method
  • convergence rate
  • diffusion approximation
  • generator comparison
  • prelimit

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Statistics, Probability and Uncertainty
  • Management Science and Operations Research

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