A Poisson approximation for the Dirichlet law, the Ewens sampling formula and the Griffiths-Engen-McCloskey law by the Stein-Chen coupling method

Abstract

We consider the random number of (Griffiths-Engen-McCloskey (GEM))-(Poisson-Dirichlet) components which are greater than ε. In two alternative and similar ways, letting Dirichlet laws and Ewens sampling formula laws respectively converge to the GEM-(Poisson-Dirichlet) law and using the Stein-Chen coupling method, we prove the Poisson approximation with respect to the total variation metric of the satisfactory order of magnitude 1/expectation.