Poisson-Dirichlet approximation for the stationary distribution of the inclusion process

We consider the approximation of the stationary distribution of the finite inclusion process with the Poisson-Dirichlet distribution. Using Stein’s method, we derive an explicit bound for the approximation error, which is of order 1/N in the thermodynamic limit. The results are achieved from a minor modification to Stein’s method for Poisson-Dirichlet distribution approximation developed in Gan & Ross (2021). The derivatives used on test functions in Gan & Ross (2021) were directional type derivatives specifically chosen for their measure preserving properties. Depending upon the application, these derivatives can prove cumbersome. In this note, we show that for certain test functions we can instead use more traditional derivatives, which simplifies the bounds for the Stein factors and is more amenable to the approximation of the inclusion process.

💡 Research Summary

The paper investigates the stationary distribution of the finite inclusion process—a stochastic interacting particle system—and shows that it can be accurately approximated by a Poisson‑Dirichlet (PD) distribution using Stein’s method. The inclusion process consists of N particles distributed over L lattice sites; particles jump between sites at rates proportional to the product of their occupation fractions and undergo independent “mutation” events at rate θ/L, making the model equivalent to a Moran‑type population genetics model. Existing literature (e.g., Gan & Ross 2021) developed a Stein‑type framework for approximating Dirichlet processes and PD distributions, but relied on a directional derivative operator (∂ₓF(μ)=lim_{ε→0+}