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 and Ross, (2021). The derivatives used on test functions in Gan and 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.