Poisson processes in the space of \((d-1)\)-dimensional totally geodesic subspaces (hyperplanes) in a d-dimensional hyperbolic space of constant curvature \(-1\) are studied. The k-dimensional Hausdorff measure of their k-skeleton is considered. Explicit formulas for first- and second-order quantities restricted to bounded observation windows are obtained. The central limit problem for the k-dimensional Hausdorff measure of the k-skeleton is approached in two different set-ups: (i) for a fixed window and growing intensities, and (ii) for fixed intensity and growing spherical windows. While in case (i) the central limit theorem is valid for all \(d\ge 2\), it is shown that in case (ii) the central limit theorem holds for \(d\in \{2,3\}\) and fails if \(d\ge 4\) and \(k=d-1\) or if \(d\ge 7\) and for general k. Also rates of convergence are studied and multivariate central limit theorems are obtained. Moreover, the situation in which the intensity and the spherical window are growing simultaneously is discussed. In the background are the Malliavin–Stein method for normal approximation and the combinatorial moment structure of Poisson U-statistics as well as tools from hyperbolic integral geometry.

泊松在的空间处理\（（d-1）\）在维全测子空间（超平面）d恒定曲率的维双曲空间（ - 1 \）\进行了研究。考虑其k骨架的k维Hausdorff度量。获得了限制在有界观察窗中的一阶和二阶量的显式。k骨架的k维Hausdorff测度的中心极限问题在两种不同的设置中得到解决：（i）固定窗口和强度增长，（ii）固定强度和球面窗口增长。在（i）情况下，中心极限定理对所有\（d \ ge 2 \），表明在情况（ii）中，中心极限定理适用于\（d \ in \ {2,3 \} \），如果\（d \ ge 4 \）和\（k = d-1 \）或\（d \ ge 7 \）以及一般k。还研究了收敛速度，并获得了多元中心极限定理。此外，讨论了强度和球形窗口同时增长的情况。在背景中是用于法线逼近的Malliavin–Stein方法和Poisson U统计量的组合矩结构以及双曲积分几何中的工具。