Statistical Bergman geometry
This paper explores the Bergman geometry of bounded domains $Ω$ in $\mathbb{C}^n$ through the lens of information geometry by introducing a mapping $Φ: Ω\rightarrow \mathcal{P}(Ω)$, where $\mathcal{P}(Ω)$ denotes a space of probability measures on $Ω$. A result by J. Burbea and C. Rao establishes that the pullback of the Fisher information metric, the fundamental Riemannian pseudo-metric in information geometry, via $Φ$ coincides with the Bergman metric of $Ω$. Building on this idea, we consider $Ω$ as a statistical model and present several interesting results within this framework. First, we derive a new statistical curvature formula for the Bergman metric by expressing it in terms of covariance. Second, given a proper holomorphic map $f: Ω_1 \rightarrow Ω_2$, we prove that if the induced measure push-forward $κ: \mathcal{P}(Ω_1) \rightarrow \mathcal{P}(Ω_2)$ preserves the Fisher information metrics, then $f$ must be a biholomorphism. Finally, we establish the consistency and the central limit theorem of the Fréchet sample mean for Calabi's diastasis function.