# Error bounds for the normal approximation to the length of a Ewens partition

## TL;DR

This paper derives error bounds for the normal approximation of the Ewens partition length, revealing how the approximation accuracy varies across different asymptotic regimes as parameters grow large.

## Contribution

It provides the first explicit error bounds for the normal approximation of Ewens partition lengths under various asymptotic conditions.

## Key findings

- Error bounds depend on the asymptotic regime.
- Normal approximation improves as n and θ grow large.
- Decay rate of error varies with the relationship between n and θ.

## Abstract

Let $K(=K_{n,\theta})$ be a positive integer-valued random variable whose distribution is given by ${\rm P}(K = x) = \bar{s}(n,x) \theta^x/(\theta)_n$ $(x=1,\ldots,n) $, where $\theta$ is a positive number, $n$ is a positive integer, $(\theta)_n=\theta(\theta+1)\cdots(\theta+n-1)$ and $\bar{s}(n,x)$ is the coefficient of $\theta^x$ in $(\theta)_n$ for $x=1,\ldots,n$. This formula describes the distribution of the length of a Ewens partition, which is a standard model of random partitions. As $n$ tends to infinity, $K$ asymptotically follows a normal distribution. Moreover, as $n$ and $\theta$ simultaneously tend to infinity, if $n^2/\theta\to\infty$, $K$ also asymptotically follows a normal distribution. In this paper, error bounds for the normal approximation are provided. The result shows that the decay rate of the error changes due to asymptotic regimes.