Limit Theorems for the Pitman-Yor Frequency Spectrum

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 $\sum _{j=\lf \lambda n\rf}^{\lf \mu n\rf} M_{jn}$, $0<\lambda\le \mu\le 1$, are obtained. Our results suggest a possible functional limit theorem for $\sum _{j=\lf \lambda n\rf}^{n} M_{jn}$. Useful connections with limit shapes for random structures on the set of partitions and other applications are suggested.