A Law of large numbers for vector-valued linear statistics of Bergman DPP

2024/04/23 by Zhaofeng Lin, Lin, Zhaofeng, Yanqi Qiu +3
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Functional Analysis (math.FA) #Geometry and complex manifolds #Holomorphic and Operator Theory #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2404.14978

openalex publication_date 2024/04/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish a law of large numbers for a certain class of vector-valued linear statistics for the Bergman determinantal point process on the unit disk. Our result seems to be the first LLN for vector-valued linear statistics in the setting of determinantal point processes. As an application, we prove that, for almost all configurations X with respect to with respect to the Bergman determinantal point process, the weighted Poincaré series (we denote by d_h(⋅,⋅) the hyperbolic distance on \mathbbD) \beginalign* ∑_k=0^∞∑_x∈ X\atop k≤ d_h(z,x)

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