Mode and Edgeworth expansion for the Ewens distribution and the Stirling numbers

Abstract

We provide asymptotic expansions for the Stirling numbers of the first kind and, more generally, the Ewens (or Karamata-Stirling) distribution. Based on these expansions, we obtain some new results on the asymptotic properties of the mode and the maximum of the Stirling numbers and the Ewens distribution. For arbitrary $\theta>0$ and for all sufficiently large $n\in\mathbb N$, the unique maximum of the Ewens probability mass function $$ \mathbb L_n(k) = \frac{\theta^k}{\theta(\theta+1)\ldots(\theta+n-1)} \genfrac{[}{]}{0pt}{}{n}{k}, \quad k=1,\ldots,n, $$ is attained at $k= \left\lfloor \theta \log n + \frac{\theta \Gamma'(\theta)}{\Gamma(\theta)} - \frac 12\right\rfloor$ or $k=\left\lceil \theta \log n + \frac{\theta \Gamma'(\theta)}{\Gamma(\theta)} + \frac 12\right\rceil$. We prove that the mode is $$ k=\left\lfloor \theta\log n - \frac{\theta \Gamma'(\theta)}{\Gamma(\theta)}\right\rfloor $$ for a set of $n$'s of asymptotic density $1$, yet this formula is not true for infinitely many $n$'s.