On the Rate of Convergence of Mean-Field Models: Stein's Method Meets the Perturbation Theory

This paper studies the rate of convergence of a family of continuous-time Markov chains (CTMC) to a mean-field model. When the mean-field model is a finite-dimensional dynamical system with a unique equilibrium point, an analysis based on Stein’s method and the perturbation theory shows that under some mild conditions, the stationary distributions of CTMCs converge (in the mean-square sense) to the equilibrium point of the mean-field model if the mean-field model is globally asymptotically stable and locally exponentially stable. In particular, the mean square difference between the Mth CTMC in the steady state and the equilibrium point of the mean-field system is O(1/M), where M is the size of the Mth CTMC. This approach based on Stein’s method provides a new framework for studying the convergence of CTMCs to their mean-field limit by mainly looking into the stability of the meanfield model, which is a deterministic system and is often easier to analyze than the CTMCs. More importantly, this approach quantifies the rate of convergence, which reveals the approximation error of using mean-field models for approximating finite-size systems.