Evaluating moments of length of Pitman partition

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 $\mathsf{E}[K^r]$ ($r=1,2,\dots$) 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 $K$.

arXiv comment10 pages. Added a reference and corrected errors

References in corpus (3)

• Large deviations associated with Poisson--Dirichlet distribution and Ewens sampling formula
• A Berry-Esseen theorem for Pitman's $\alpha$-diversity
• Error bounds for the normal approximation to the length of a Ewens partition