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

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 Poincar\'e series (we denote by dₕ(·,·) the hyperbolic distance on D) {align*} ∑ₖ₌₀^∞∑_{x∈ X k≤ dₕ(z,x)<k+1}e^{-sdₕ(z,x)}f(x) {align*} cannot be simultaneously convergent for all Bergman functions f∈ A²(D) whenever $1<s<3/2$. This confirms a result announced without proof in Bufetov-Qiu's work.