We derive large-sample and other limiting distributions of the ``frequency of frequencies'' vector, ${\bf M_n}$, together with the number of species, $K_n$, in a Poisson-Dirichlet or generalised Poisson-Dirichlet gene or species sampling model. Models analysed include those constructed from gamma and $\alpha$-stable subordinators by Kingman, the two-parameter extension by Pitman and Yor, and another two-parameter version constructed by omitting large jumps from an $\alpha$-stable subordinator. In the Poisson-Dirichlet case ${\bf M_n}$ and $K_n$ turn out to be asymptotically independent, and notable, especially for statistical applications, is that in other cases the conditional limiting distribution of ${\bf M_n}$, given $K_n$, is normal, after certain centering and norming.

翻译：我们推导了泊松-狄利克雷或广义泊松-狄利克雷基因/物种抽样模型中“频率之频率”向量 ${\bf M_n}$ 以及物种数 $K_n$ 的大样本及其他极限分布。所分析的模型包括：Kingman 基于伽马和 $\alpha$-稳定子序构造的模型、Pitman 与 Yor 提出的双参数扩展模型，以及通过从 $\alpha$-稳定子序中剔除大跳跃构造的另一双参数版本。在泊松-狄利克雷情形下，${\bf M_n}$ 与 $K_n$ 渐近独立；值得注意的是（尤其对统计应用而言），在其他情形下，经过特定中心化与标准化后，给定 $K_n$ 时 ${\bf M_n}$ 的条件极限分布为正态分布。