Evaluating moments of length of Pitman partition

TL;DR

This paper analyzes the asymptotic behavior of the moments of the length of partitions generated by the Pitman sampling formula, providing refined approximations under various asymptotic regimes.

Contribution

It offers new asymptotic evaluations of the moments of the partition length, improving accuracy over previous results and considering simultaneous growth of parameters.

Findings

Asymptotic formulas for the first and higher moments of K.

Refined approximations as n→∞.

Results for parameters with θ/n→0.

Abstract

The Pitman sampling formula has been intensively studied as a distribution of random partitions. One of the objects of interest is the length $K (= K_{n,\theta,\alpha})$ of a random partition that follows the Pitman sampling formula, where $n\in\mathbb{N}$, $\alpha\in(0,\infty)$ and $\theta > -\alpha$ are parameters. This paper presents asymptotic evaluations of its $r$-th moment $\mathsf{E}[K^r]$ ($r=1,2,\ldots$) under two asymptotic regimes. In particular, the goals of this study are to provide a finer approximate evaluation of $\mathsf{E}[K^r]$ as $n\to\infty$ than has previously been developed and to provide an approximate evaluation of $\mathsf{E}[K^r]$ as the parameters $n$ and $\theta$ simultaneously tend to infinity with $\theta/n \to 0$. The results presented in this paper will provide a more accurate understanding of the asymptotic behavior of $K$.