The Prelimit Generator Comparison Approach of Stein’s Method

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.

The join-the-shortest-queue system in the Halfin-Whitt regime: rates of convergence to the diffusion limit

“…where G X is the CTMC generator. For a proof, we refer readers to Lemma 2 of Braverman (2021); see also Lemma 1 of Barbour (1988). Since the right-hand side is independent of c, we fix c = −f (0) h (x q ) for convenience and define…”

“…Techniques proposed to obtain multidimensional Stein factor bounds include using a priori Schauder estimates from elliptic PDE theory as in Gurvich (2014), using couplings to analyze and bound the sensitivity of the diffusion to its initial condition as in Barbour (1988) and Mackey and Gorham (2016), and bounding the Stein factors using Malliavin calculus as in Fang et al (2018) and Jin et al (2021). A detailed description of these techniques can be found in Section 1.1 of Braverman (2021). However, despite progress on multidimensional Stein factor bounds, the JSQ system is not covered by existing results because our limiting diffusion in (1) is constrained to the nonnegative orthant via reflecting boundary conditions.…”

“…To deal with the Stein factor bound problem, this paper promotes the use of the prelimit generator comparison approach, which was recently proposed by Braverman (2021) as an alternative to the generator comparison approach. The prelimit approach is the mirror image of the classical generator approach.…”

“…For the moment bounds used in this paper, the result that all moments of Y are finite, proved by Banerjee and Mukherjee (2019), is sufficient because our limit Y does not depend on n. The Stein factor bounds pose a bigger challenge, and we deal with them in Section 3. It was noted in Braverman (2021) that the prelimit and classical generator comparison approaches should be equivalent, in theory, in the sense that any bound on |Eh(X) − Eh(Y )| obtained using one of them should, in theory, be attainable using the other. However, in practice, one approach could be more tractable, or convenient, to work with; see, for instance, the example in Section 4…”

“…of Braverman (2021). In our own paper, the discrete state space of the JSQ system greatly simplifies tracking the different couplings used in Section 3.2.…”