Poisson-Dirichlet approximation for the stationary distribution of the inclusion process
Han L. Gan
TL;DR
This work develops a refined Stein's-method framework to approximate the stationary distribution of the finite inclusion process by the Poisson-Dirichlet/Dirichlet-process model DP(θ,π). It introduces standard-derivative bounds via a Fleming–Viot dual, enabling explicit, computable bounds on the Stein equation solutions and a concrete generator-comparison argument. For the discrete inclusion process, it proves an explicit O(1/N) bound on the distributional distance to DP(θ,π) and establishes sampling-formula convergence with the Ewens formula, with thermodynamic-limit scaling N,L → ∞ and N/L → θ. These results provide precise error controls and broaden the applicability of PD/DP approximations to interacting particle systems.
Abstract
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.