🤖 AI Summary
This study investigates the large-sample asymptotic behavior of the frequency spectrum in Pitman–Yor random partitions, with particular emphasis on the limiting distribution of the sum of frequency counts over intervals of the form ∑_{j=⌊λn⌋}^{⌊μn⌋} M_{jn}. By leveraging the theory of Gibbs-type partitions, asymptotic analysis, and methods from combinatorial stochastic structures, the work establishes, for the first time, a limit theorem for the Pitman–Yor frequency spectrum that holds uniformly across broad ranges of such intervals, and further explores its functional limit form. The results uncover a profound connection between the frequency spectrum and the limiting shape of associated random combinatorial structures, thereby providing a rigorous theoretical foundation for applications in population genetics, particularly in the analysis of allele frequency spectra.
📝 Abstract
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.