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

Abstract

Let $K(=K_{n,\theta })$ be a positive integer-valued random variable whose distribution is given by $\mathrm{P}(K=x)=\bar{s}(n,x){\theta }^x/(\theta {)}_n$ $(x=1,\dots ,n)$, 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 $x=1,\dots ,n$. 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.

arXiv comment18 pages

References in corpus (1)

• Large deviations associated with Poisson--Dirichlet distribution and Ewens sampling formula

Cited by in corpus (1)

• Evaluating moments of length of Pitman partition