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

TL;DR

This paper derives asymptotic expansions for Stirling numbers and the Ewens distribution, revealing new insights into their modes and maxima for large n, with precise formulas and exceptions.

Contribution

It provides novel asymptotic formulas for the mode and maximum of the Ewens distribution and Stirling numbers, including conditions for their accuracy.

Findings

Asymptotic expansion formulas for Stirling numbers and Ewens distribution.

Precise characterization of the mode for large n.

Identification of exceptions where the mode formula does not hold.

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.