Consider a Poisson point process within a convex set in a Euclidean space. The Vietoris-Rips complex is the clique complex over the graph connecting all pairs of points with distance at most $\delta$. Summing powers of the volume of all $k$-dimensional faces defines the volume-power functionals of these random simplicial complexes. The asymptotic behavior of the volume-power functionals of the Vietoris-Rips complex is investigated as the intensity of the underlying Poisson point process tends to infinity and the distance parameter goes to zero. Univariate and multivariate central limit theorems are proven. Analogous results for the \v{C}ech complex are given.

中文翻译：

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随机单纯复形的多元中心极限定理

考虑欧几里得空间中凸集内的泊松点过程。Vietoris-Rips 复合体是连接所有距离最大为 $\delta$ 的点对的图上的集团复合体。所有 $k$ 维面的体积幂求和定义了这些随机单纯复形的体积幂函数。随着潜在泊松点过程的强度趋于无穷大且距离参数趋于零，研究了 Vietoris-Rips 复合体的体积功率泛函的渐近行为。证明了单变量和多变量中心极限定理。给出了 \v{C}ech 复合体的类似结果。