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

TL;DR

This paper develops a Stein's method-based approach to approximate the stationary distribution of the finite inclusion process using the Poisson-Dirichlet distribution, providing explicit error bounds and simplifying derivative calculations.

Contribution

It introduces a modified Stein's method with traditional derivatives for Poisson-Dirichlet approximation, improving simplicity and applicability for the inclusion process.

Findings

Explicit 1/N error bound for the approximation

Modified Stein's method with traditional derivatives

Simplified bounds for test functions

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.