Mode and Edgeworth Expansion for the Ewens Distribution and the Stirling Numbers

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 $\theta0$ and for all sufficiently large $n\in\mathbb{N}$ , the unique maximum of the Ewens probability mass function

$\begin{displaymath}\mathbb L_n(k) = \frac{\theta^k}{\theta(\theta+1)\cdots(\theta+n-1)} \genfrac{[}{]}{0pt}{}{n}{k}, \quad k=1,\ldots,n, \end{displaymath}$

is attained at $k= \lfloor a_n \rfloor$ or $\lceil a_n \rceil$ , where $a_n = \theta \log n - \theta \Gamma'(\theta)/\Gamma(\theta) -1/2$ . We prove that the mode is the nearest integer to a_n for a set of n's of asymptotic density 1, yet this formula is not true for infinitely many n's.