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 $d_{h}(\cdot,\cdot)$ the hyperbolic distance on $\mathbb{D}$) \begin{align*} \sum_{k=0}^\infty\sum_{x\in X\atop k\le d_{h}(z,x)<k+1}e^{-sd_{\mathrm{h}}(z,x)}f(x) \end{align*} cannot be simultaneously convergent for all Bergman functions $f\in A^2(\mathbb{D})$ whenever $1<s<3/2$. This confirms a result announced without proof in Bufetov-Qiu's work.