Poisson Approximations for Sum of Bernoulli Random Variables and its Application to Ewens Sampling Formula

The Ewens sampling formula is well-known as a distribution of a random partition of the set of integers {1, 2, . . . , n}. We give the condition that the number Kn of distinct components of the formula converges to the shifted Poisson distribution. Based on this convergence, we give the new approximations to the distribution of Kn, which are different from the approximations by Arratia et al. (2000, 2003). The formers are better than the latters. This is shown by comparing the bounds for the total variation distances between the distributions of the approximations and the distribution of Kn. Several examples are given to illustrate the results.