Limit Theorems for the Pitman-Yor Frequency Spectrum

2026/07/26 by Ross Arthur Maller, Soudabeh Shemehsavar
Mathematics · #math.PR #math.ST #stat.ME #stat.TH

paper · pdf

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≤ j≤ n} (in genetics, the allele frequency spectrum) associated with a random partitioning of \1,2,…, n\. The two-parameter Pitman-Yor sampling model is analysed in detail and asymptotic distributions of sums of the form ∑ _{j=\lf λn\rf}^{\lf μn\rf} M_jn, 0<λ≤ μ≤ 1, are obtained. Our results suggest a possible functional limit theorem for ∑ _{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.

Citations

• Poisson Hierarchical Indian Buffet Processes-With Indications for Microbiome Species Sampling Models
• Random Discrete Probability Measures Based on Negative Binomial Process
• Inverse clustering of Gibbs Partitions via independent fragmentation and dual dependent coagulation operators
• Gibbs partitions: a comprehensive phase diagram
• Gibbs Partitions, Riemann-Liouville Fractional Operators, Mittag-Leffler Functions, and Fragmentations Derived From Stable Subordinators
• Negative Binomial Construction of Random Discrete Distributions on the Infinite Simplex
• Convergence to stable limits for ratios of trimmed Levy processes and\n their jumps
• The limit distribution of ratios of jumps and sums of jumps of subordinators
• The generalized weighted probability measure on the symmetric group and the asymptotic behavior of the cycles
• Ratio theorems for random walks II

Related

• Limit Theorems Associated With The Pitman-Yor Process
• A martingale approach to Gaussian fluctuations and laws of iterated logarithm for Ewens-Pitman model
• Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent
• Empirical and Full Bayes estimation of the type of a Pitman-Yor process
• A Martingale Approach to Large-θ Ewens-Pitman Model