AI 中文总结
研究推导通用分布公式用于皮特曼 - 约尔频谱线性组合，分析两参数抽样模型得到特定形式和的渐近分布，暗示可能的泛函极限定理及与随机结构极限形状等的联系。
AI 中文摘要
我们推导了一个适用于多种吉布斯型划分的通用分布公式，并利用它得到了与\(\{1,2,\ldots, n\}\)的随机划分相关的分量频谱\((M_{jn})_{1\le j\le n}\)（在遗传学中为等位基因频谱）线性组合的大样本结果。详细分析了两参数皮特曼 - 约尔抽样模型，得到了\(\sum _{j=\lf \lambda n\rf}^{\lf \mu n\rf} M_{jn}\)（\(0<\lambda\le \mu\le 1\)）形式和的渐近分布。结果暗示了\(\sum _{j=\lf \lambda n\rf}^{n} M_{jn}\)可能的泛函极限定理，并提出了与划分集上随机结构的极限形状及其他应用的有用联系。
英文摘要
We derive a general distribution formula applicable to a wide variety of Gibbs-type partitions and use it to obtain large sample results for linear combinations of the component frequency spectrum $(M_{jn})_{1\le j\le n}$ (in genetics, the allele frequency spectrum) associated with a random partitioning of $\{1,2,\ldots, n\}$. The two-parameter Pitman-Yor sampling model is analysed in detail and asymptotic distributions of sums of the form $\sum _{j=\lf λn\rf}^{\lf μn\rf} M_{jn}$, $0<λ\le μ\le 1$, are obtained. Our results suggest a possible functional limit theorem for $\sum _{j=\lf λn\rf}^{n} M_{jn}$. Useful connections with limit shapes for random structures on the set of partitions and other applications are suggested.
