The prelimit generator comparison approach of Stein's method

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 establish derivative bounds for the solution to the Poisson equation, known as Stein factor bounds. In this paper we propose starting with the Markov chain Poisson equation; we term this the prelimit approach. Although Stein factor bounds still must be established, they now correspond to the finite differences of the Markov chain Poisson equation solution rather than the derivatives of the solution to the diffusion Poisson equation. In certain cases, finite difference bounds are easier to obtain for example, when the drift of the diffusion is not everywhere differentiable, or in the presence of a reflecting boundary condition. We use the $M/M/1$ model as a simple working example to illustrate our approach. In a companion paper, we apply the prelimit approach to the join-the-shortest-queue model, which is a considerably more involved multidimensional model with a state-space collapse component.